OL/L  ,e_  v_ 


THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

LOS  ANGELES 


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SOUTHERN  BRANCH, 

iJNIVERSITY  OF  CALIFORNIA, 

LIBRARY, 

ILOS  ANGELES.  CALIF. 


0.  1^  ^^ 


A><.^t  ^ 


ELEMENTS 


OF 


QUATERNION^S. 


BY 


A.    S.    HAEDY,    Ph.D., 

PROFESSOR   OF   MATHEJIATICS,    DARTMOUTH    COLLEGE. 


45043 


BOSTON: 

PUBLISHED  BY  GINN  &  COMPANY. 

188  7. 


Entered  according  to  Act  of  Congress,  in  the  year  1881,  by 

A.  S.  HARDY, 
in  the  office  of  the  Librarian  of  Congress,  at  Washington. 


J.  S.  Gushing  &  Co.,  Printers,  Boston. 


i 


Engineering  & 
Mathematical 
Sciences 
•  Library 

a  A 

PREFACE. 


rpHE  object  of  the  following  treatise  is  to  exhibit  the 

elementary  principles  and  notation  of  the  Quaternion 

Calculus,  so  as  to  meet  the  wants  of  beginners  in  the 

class-room.     The  Elements  and  Lectures  of  Sir  William 

Rowan  Hamilton,  while  they  may  be  said  to  contain  the 

^     suggestion  of  all  that  will  be  done  in  the  way  of  Quater- 

\      nion  research  and  application,  are  not,  for  this  reason,  as 

also  on  account  of  their  cliffuseness  of  style,  suitable  for 

the  purposes  of  elementary  instruction.     Tait's  work  on 

Quaternions   is   also,  in   its   originality   and   conciseness, 

beyond  the  time  and  needs  of  the  beginner.     In  addition 

to  the  above,  the  foUowmg  works  have  been  consulted: 

6^  Calcolo  del  Quaternione.     Bellavitis;  Modena,  1858. 

Exposition  de  la  Metliode  des  Equipollences.  Traduit 
de  ritalien  de  Giusto  Bellavitis,  par  C.-A.  Laisant ;  Paris, 
1874.  (Original  memoir  in  the  Memoirs  of  the  Italian 
Society.     1854.) 

Theorie  Elementaire  des  Quantites  Complexes.  J. 
Hoiiel;    Paris,  1874. 

Essai  sur  une  Mani^re  de  Representer  les  Quantites 
Imaginaires  dans  les  Construction  GSometriques.  Par 
R.  Argand ;    Paris,  1806.      Second  edition,  with  j^reface 


IV  PREFACE. 

l)y  J.  Iloiiel;  Paris,  1874.  Translated,  with  notes,  from 
the  French,  by  A.  S.  Hardy.  Van  Nostrand's  Science 
Series,  No.  52;    1881. 

Kurze  Anleitung  zum  Rechnen  mit  den  (^Hamilton' sclieri) 
Qunternionen.     J.  Odstrcil ;    Halle,  1879. 

Applications  Mecaniques  du  Calcul  des  Quaternions. 
Laisant;   Paris,  1877. 

Introduction  to  Quaternions.  KeUand  and  Tait;  Lon- 
don, 1873. 

A  free  use  has  been  made  of  the  examples  and  exercises 
of  the  last  work ;  and,  in  Article  87,  is  given,  by  permis- 
sion, the  substance  of  a  paper  from  Volume  I.,  page  379, 
American  Journal  of  Mathematics,  illustrating  admirably 
the  simplicity  and  brevity  of  the  Quaternion  method. 

If  this  presentation  of  the  principles  shall  afford  the 
undergraduate  student  a  glimpse  of  this  elegant  and  pow- 
erful instrument  of  analytical  research,  or  lead  him  to 
follow  their  more  extended  api)lication  in  the  works  above 
cited,  the  aim  of  this  treatise  will  have  been  accomplished. 

The  author  expresses  his  obligation  to  Mr.  T.  W.  D. 
Worthen  for  valuable  assistance  in  the  prejDaration  of 
this  work,  and  to  Mr.  J.  S.  Gushing  for  whatever  of 
typographical  excellence  it  possesses. 

A.  S.  HARDY. 
Hanover,  N.H.,  June  21,  1881. 


COJSTTEI^TS. 


CHAPTER  I. 

Addition    and    Subtraction   of  Vectors ;    or,   Geometric 
Addition  and  Subtraction. 

Article.  Page. 

1.  Definition  of  a  vector.     Effect  of  the  minus  sign  before  a 

vector .  1 

2.  Equal  vectors      2 

3.  Unequal  vectors.     Vector  addition 2 

4.  Vector  addition,  commutative 3 

6.  Vector  addition,  associative      3 

6.  Transposition  of  terms  in  a  vector  equation 4 

7.  Definition  of  a  tensor 4 

8.  Definition  of  a  scalar 5 

9.  Distributive  law  in  the  multiplication  of  vector  by  scalar 

quantities 6 

10.  If  2a  +  2,3  =  0,  then  Sa  =  0  and  2,3  =  0 7 

11.  Examples 8 

12.  Complanar  vectors.     Condition  of  complanarity 15 

13.  Co-initial  vectors.     Condition  of  coUinearity      16 

14.  Examples 17 

15.  Expression  for  a  medial  vector 24 

16.  Expression  for  an  angle-bisector 25 

17.  Examples 26 

18.  Mean  point      28 

19.  Examples 28 

20.  Exercises 30 

CPIAPTER  II. 

Multiplication  and  Division  of  Vectors ;    or,   Geometric 
Multiplication  and  Division. 

21.  Elements  of  a  quaternion 32 

22.  Equal  quaternions 34 

23.  Positive  rotation 35 


n  CONTENTS. 

Article.  Page. 

24.  Analytical  expression  for  a  quaternion.     Product  and  quo- 

tient of  rectangular  unit-vectors.     Tensor  and  versor  of  a 

quaternion 36 

25.  Symbolic  notation  q  =  TqVq 39 

26.  Reciprocal  of  a  quaternion 39 

27.  Quadrantal  versors,  i,  j,  k 40 

28.  Whole  powers  of  unit  vectors.    Square  of  a  unit  vector  is —  1,  41 

29.  Associative  law  in  the  multiplication  of  rectangular  unit- 

vectors     42 

30.  Negative  sign  commutative  with  i,  j,  k 43 

31.  Commutative  law  not  true  for  the  products  of  i,  j,  k   .   .   .   .  43 

32.  Reciprocal  of  a  unit  vector 44 

33.  A  unit  vector  commutative  with  its  reciprocal.     Reciprocal 

of  any  vector 45 

34.  Product  and  quotient  of  any  two  rectangular  vectors  ....  47 

35.  Square  of  any  vector 47 

36.  Distributive  law  true  of  the  products  of  i.  j.  k 48 

37.  Exercises 1 48 

38.  Symbolic  notation,  q  =  Sq  +  Yq 49 

39.  De  Moivre's  theorem 50 

40.  Products  of  two  vectors.     Symbolic  notation 56 

41.  General  principles  and  formulae 58 

42.  Powers  of  vectors  and  quaternions Gl 

43.  Relation  between  the  vector  and  Cartesian  determination  of 

a  point G4 

44.  Right,  complanar.  diplanar  and  collinear  quaternions.     Any 

two  quaternions  reducible  to  the  forms  -,  1  and  -,  1  .   .  65 

/3   /?  yd 

45.  Reciprocal  of  a  vector,  scalar  and  quaternion 66 

46.  Conjugate  of  a  vector,  scalar  and  quaternion 68 

47.  Opposite  quaternions 70 

48.  Vl\.q  =  V-  =  —  =  KVq :    .   .  71 

q      Vq 

49.  Representation  of  versors  by  spherical  arcs 71 

60.   Addition  and  subtraction  of  quaternions.     K,  S  and  V  dis- 
tributive symbols.     K  commutative  with  S  and  V.     T  and 

U  not  distributive  symbols 72 

51.  Multiplication  of  quaternions;  not  commutative.  \jUq  =  U\]q. 

TTlq  =  X\Tq.    K(/r=  KrK'/.     Product  or  quotient  of  com- 
planar quaternions 74 

52.  Distributive  and  associative  laws  in  quaternion  and  vector 

multiplication 79 

53.  General  formulae 82 


CONTENTS.  Til 

Article.  Page. 

54.  Applications 82 

55.  rormulae  relating  to  the  products  of  two  or  more  vectors  .    .  99 

56.  Exercises 106 

57.  Examples.     Applications  to  Spherical  Trigonometry   ....  108 

58.  General  Formulae 119 

59.  Applications  to  Plane  Trigonometry 125 


CHAPTER  III. 
Applications    to    Loci. 

60.  General  equations  of  a  line  and  surfiice 131 

61.  Use  of  Cartesian  forms  in  conjunction  with  quaternion  rea- 

soning   •  .  132 

62.  Non-commutative  law  in  quaternion  diflferentiation.     Differ- 

entiation of  scalar  fimctious 132 

63.  Quaternion  differentiation 134 

64.  Illustratiou 13G 

65.  Distributive  principle 138 

66.  Quadrinomial  form 139 

67.  Examples 140 

The  Right  Line. 

68.  Right  line  through  the  origin 145 

69.  Parallel  lines 145 

70.  Right  line  through  two  given  points 146 

71.  Perpendicular  to  a  given  line ;  its  length 147 

72.  Equations  of  a  right  line  are  linear  and  involve  one  indepen- 

dent variable  scalar 150 

The  Plane. 

73.  Equation  of  a  plane 151 

74.  Plane  making  equal  angles  with  three  given  lines      152 

75.  Plane  through  three  given  points 153 

76.  Equations  of  a  plane  are  linear  and  involve  two  independent 

variable  scalars      154 

77.  Exercises  and  problems  on  the  right  line  and  plane 154 

The  Circle  and  Sphere. 

78.  Equations  of  the  circle 164 

79.  Equations  of  the  sphere 165 


viii  CONTENTS. 

Article.  Pi(W. 

80.  Tangent  line  and  plane 1(!G 

81.  Chords  of  contact 1G7 

82.  Exercises  and  problems  on  the  circle  and  the  sphere    ....  167 

83.  Exercises  in  the  transformation  and  interpretation  of  ele- 

meutaiy  symbolic  forms     176 

The  Conic  Sections.      Cartesian  Forms. 

84.  The  parabola 178 

85.  Tangent  to  the  parabola     ,    .    .    .  178 

86.  Examples  on  the  parabola 180 

87.  Relations  between  three  intersecting  tangents  to  the  para- 

bola    185 

88.  The  ellipse 101 

89.  Examples  on  the  ellipse 102 

90.  The  hj'perbola 105 

91.  Examples  on  the  h3'perbola    ....       105 


92.  Linear  equation  in  quaternions.     Conjugate  and  self-conju- 

gate functions 100 

93.  General  equations  of  the  conic  sections 201 

94.  The  ellipse 204 

95.  Examples 206 

96.  The  parabola 2U 

97.  Examples 216 

98.  The  cycloid 222 

99.  Elementarj'  applications  to  mechanics 223 

100.  Miscellaneous  Examples 231 


ELEMENTS  OF  QUATERNIONS. 


QUATERNIOI^S. 


CHAPTER   I. 

Addition  and  Subtraction  of  Vectors,  or  Geometric  Addition  and 
Subtraction. 

1.  A  Vector  is  the  representative  of  transference  throvgh  a 
given  distance  in  a  given  direction. 

Thus,  if  A,  B  are  anj'-  two  points,  vector  ab  implies  a  trans- 
lation from  A  to  B. 

A  vectd"  ma}"  be  represented  geometricall}'  b}-  a  right  line, 
whose  length  denotes  the  distance  over  which  transference  takes 
place,  and  whose  dii-ection  denotes  the  direction  of  the  trans- 
ference. In  thus  designating  a  rector,  the  direction  is  indicated 
by  the  order  of  the  letters. 

Thus,  AB  (Fig.  1)  denotes  transference      ^^^-  ^- 

from  A  to  B,  and  ba  from  b  to  a. 

Retaining  the  algebraic  signification  of  the  signs  +  and  — ,  if 
AB  denotes  motion  from  a  to  b,  then  —  ab  will  denote  motion 
from  B  to  A,  and 

AB=— BA,        —  AB  =  BA  ....       (1). 

Hence,  the  effect  of  a  minus  sign  before  a  vector  is  to  reverse 
its  direction. 

The  conception  of  a  vector,  therefore,  implies  that  of  its  two 
elements,  distance  and  direction;  it  was  fii'st  defined  as  a  directed 
right  line.  It  is  now  applied  more  generally  to  all  quantities 
determined  by  magnitude  and  direction.     Thus,  force,  the  path 


QUATERNIONS. 


of  a  moving  bod}',  velocity,  an  electric  cun-eut,  etc.,  are  vector 
quantities. 

Anal^'ticall}-,  vectors   arc  represented  by  the  letters  of  the 
Greek  alphabet,  a,  p,  y,  etc. 

2.   It  follows,  from  the  definition  of  a  vector,  that  all  lines 
u'hich  are  equal  and  parallel  may  he  represented  by  the  same  vec- 
tor symbol  loith  like  or  tinlike  signs. 
If  equal  and  drawn  in  the  same 
direction,  they  will  have  the  same 
sign.     Hence  an  equality  between 
two  vectors  implies  equalit}'  in  dis- 
tance with  the  same  direction. 
Thus,  if  AB  (Fig.  2),  cd,  be,  ef 
and  iiG  are  equal  and  drawn  in  the  same  direction,  thc^'  ma}'  be 
represented  by  the  same  vector  symbol,  and 

AB  =  CD  =  BE  =  EF  =  no  =  a      .       .       .       .       (2)  . 


3.  It  follows  also  from  the  definition  of  a  A'cctor  that,  if  vec- 
tors are  not  parallel,  the}-  cannot  be  represented  b}'  the  same 
vector  symbol. 

Thus,  if  the  point  a  (Fig.  3)  move  over  the  right  line  ab, 
from  A  to  b,  and  then  over  the  right  line  bc,  from  b  to  c,  and 

AB  =  a,  BC  must  be  denoted  by 
some  other  spubol,  as  /3. 

The  result  of  these  two  succes- 
sive translations  of  the  point  a  is 
the  same  as  that  of  the  single  and 
direct  translation  AC=y,  from  a  to 
c  ;  in  either  case  a  is  found  at  the 
extremitj'  of  the  diagonal  of  the 
parallelogram  of  which  ab  and  bc  are  the  sides.  This  combina- 
tion of  successive  translations  is  called  addition,  and  is  written 

in  the  ordinarv  wav,  ,   o  /o\ 

a-f /i  =  y (3). 

This  expression  would  be  absurd  if  the  s^-mbols  denoted  mag- 
nitudes only.     It  means  that  transference  from  a  to  b,  followed 


GEOJIETEIC    ADDITIOlSr   AND    SUBTEACTIOX.  3 

bj  transference  from  b  to  c,  is  equivalent  to  transference  from 
A  to  c.  The  sign  -|-  does  not  therefore  denote  a  numerical  ad- 
dition, or  the  sign  =  an  equality'  between  magnitudes.  It  is, 
however,  called  an  equation,  and  read,  as  usual,  "a  plus  fi  is 
equal  to  y."     This  kind  of  addition  is  called  geometric  addition. 

4.  If  the  point  a  (Fig.  3),  instead  of  moving  over  the  sides 
AB,  BC  of  the  parallelogram  abcd,  had  moved  in  succession  over 
the  other  two  sides,  ad  and  dc,  the  result  would  still  have  been 
the  same  as  that  of  the  single  translation  over  the  diagonal  ac. 
But  since  ab  and  bc  are  equal  in  length  to  dc  and  ad  respect- 
ively, and  are  drawn  in  the  same  direction,  we  have  (Art.  2) 

AB  =  DC     and     BC  =  ad, 

and  if  the  first  two  translations  are  represented  by  ab  and  bc, 
the  second  two  maj'  be  represented  hj  bc  and  ab,  or 

a  +  /3  =  /3  +  a=y (4). 

Hence  the  ox>eration  of  vector  addition  is  commutative^  or  the 
sum  of  any  number  of  given  vectors  is  independent  of  their  order. 


5.   If  the  point  a  (Fig.  4)  move  in  succession  over  the  three 
edges  AB,  bc,  cg  of  a  parallelopiped, 

Fijr.  4. 


we  have 

and 

or 


AB  -{-  BC  =  AC, 

AC  +  CG  =  AG, 

(ab  +  BC)  +  CG  =  AG. 

In  like  manner 

BC  +  CG  =  BG, 
AB  +  BG  =  AG, 


Hence 


AB  +  (bc  +  cg)  =  AG. 

(aB  +  BC)  4-  CG  =  AB  +  (bc  +  Cg)       .       .       •      (5)  , 


and  the  operation  of  vector  addition  is.  associative,  or  the  sum 
of  any  number  of  given  vectors  is  independent  of  the  mode  of 
grouping  them. 


4  QUATERNIONS. 

6.  Since,  if  AC  =  y  (Fig.  3),  then  ca=  — y,  we  have 

or,  comparing  with  equation  (3), 

a  +  /3  =  y, 

a  term  may  he  transposed  from  one  member  to  another  in  a  vector 
equation  by  changing  its  sign. 

Also,  in  every  triangle,  any  side  may  be  considered  as  the 
sum  or  difference  of  the  other  two,  depending  upon  their  direc- 
tions as  vectors.     Thus    (Fig.  3) 

y  — /3=a, 
7  -  tt  =  /?. 

It  is  to  be  obser\'ed  tliat  no  one  direction  is  assumed  as  posi- 
tive, as  in  Cartesian  Gcomotr}-.  The  only  assumption  is  that 
opposite  directions  sliall  liave  opposite  signs. '  The  results  must, 
of  course,  be  interpreted  in  accordance  with  the  primitive  as- 
sumptions.    Thus,  had  we  assumed  ba  =  a  (Fig.  3) ,  y  and  ^ 

being  as  before,  then 

p  —  «  =  y. 

a-l3=-y. 

7.  If  two  vectors  having  the  same  direction  be  added  together, 

the  sum  will  be  a  vector  in  the  same  direction.     If  the  vectors 

be  also  equal  in  length,  the  length  of  the  vector  sum  will  be  twice 

the  length  of  either.     If  n  vectors,  of  equal  length  and  drawn 

in  the  same  direction,  be  added  together,  the  sum  will  be  the 

product  of  one  of  these  vectors  by  «,  or  a  vector  having  the  same 

direction  and  whose  length  is  n  times  the  common  length.     If 

then  (Fig.  2) 

AF  =  xxn  =  XCV  =  Xa, 

where  a,  b  and  f  are  in  the  same  straight  line,  cd  =  ab,  and  x 
is  a  positive  whole  number,  x  expresses  the  ratio  of  the  lengths 
of  AF  and  a.  From  the  case  in  which  x  is  an  integer  we  pass, 
by  the  usual  reasoning,  to  that  in  which  it  is  fractional  or  in- 
commensurable. Vectors,  then,  in  the  same  direction,  have  the 
same  ratio  as  the  corresponding  lengths. 


GEOMETEIC   ADDITION  AND   SUBTEACTION.  5 

If  AB  =  a  be  assumed  as  the  uuit  vector,  then 

AF  =  ma, 

in  which  m  is  a  positive  numerical  quantity  and  is  called  the 
Tensor.  It  is  the  ratio  of  the  length  of  the  vector  ma  to  that 
of  the  unit  vector  a,  or  the  numerical  factor  by  which  the  unit 
vector  is  multiplied  to  produce  the  given  vector. 

Any  vector,  as  /?,  may  be  written  in  general  notation 

^  =  T/3U/3. 

In  this  notation,  T^S  (read  "tensor  of  y8")  is  the  numerical 
factor  which  stretches  the  unit  vector  so  that  it  shall  have  the 
proper  length  ;  hence  its  name,  tensor.  It  is,  strictly  speaking, 
an  abstract  number  without  sign,  but,  to  distinguish  between  it 
and  the  negative  of  algebra,  it  may  be  said  to  be  always  posi- 
tive. U/3  (read  "  versor  of  /?")  is  the  unit  vector  having  the 
direction  of  /3  ;  the  reason  for  the  name  versor  will  appear  later. 

T  and  U  are  also  general  symbols  of  operation.  Written  be- 
fore an  expression,  they  denote  the  operations  of  taking  the 
tensor  and  versor,  respectively.  Thus,  if  the  length  of  ji  is  n 
times  that  of  the  unit  vector, 

T(/3)  =  n, 

where  T  denotes  the  operation  of  taking  the  stretching  factor, 
i.e.  the  tensor.     "While 

r(/?)=r/3 

indicates  the  operation  of  taking  the  unit  vector,  that  is,  of 
reducing  a  vector  jB  to  its  unit  of  length  without  changing  its 
direction. 

8.   If  BC  (Fig.  5)  be  any  vector,  and  ba  =  ?/bc,  then 

—  BA  =  AB  =  —  ?/BC  ;  yjg  5^ 

and,  in  general,  if  ba  and  bc  be      5 c A 


any  two  real  vectors,  parallel  and 

of  unequal  length,  we  may  always  conceive  of  a  coefficient  y 

which  shaU  satisfy  the  equation 

BA  =  ?/BC, 


6  QUATERNIONS. 

where  y  is  plus  or  minus,  according  as  the  vectors  have  the  same 
or  opposite  directions,  y  may  bo  called  the  geometric  quotient, 
and  is  a  real  number,  plus  or  minus,  expressing  numerically  the 
ratio  of  the  vector  lengths.  This  quotient  of  2yaraUel  vectors, 
which  may  be  positive  or  negative,  whole,  fractional  or  incom- 
mensurable, but  which  is  always  real,  is  called  a  Scalar,  because 
it  may  be  always  found  by  the  actual  comparison  of  the  parallel 
vectors  with  a  parallel  right  line  as  a  scale. 

It  is  to  be  observed  that  tensors  are  pure  numbers,  or  signless 
numbers,  operating  onl}-  metrically  on  the  lengths  of  the  vectors 
of  which  the}'  are  coefficients :  while  scalars  are  sign-bearing 
numbers,  or  the  reals  of  Algebra,  and  are  combined  with  each 
other  by  the  ordinar}'  rules  of  Algebra ;  they  may  be  regarded 
as  the  product  of  tensors  and  the  signs  of  direction. 


Thus,  let 

a=  aUa. 


® 


Then  Ta  =  a.  If  we  increase  the  length  of  a  b}'  the  factor  &, 
6  is  a  tensor,  but  the  tensor  of  the  resulting  vector  is  ba.  If  we 
operate  with  —  6,  —  6  is  not  a  tensor,  for  a  is  not  onl}'  stretched 
but  also  reversed ;  the  tensor  of  the  resulting  vector  is  as  before 
ha  ;  in  other  words,  direction  does  not  enter  into  the  conception 
of  a  tensor.  As  the  product  of  a  sign  and  a  tensor,  —h  in  a 
scalar.  The  operation  of  taking  the  scalar  terms  of  an  expres- 
sion is  indicated  by  the  sj'mbol  S.  Thus,  if  c  be  an}'  real  alge- 
braic quantity, 

S  (  —  &a  Va  -\-c)  =  c, 

for  —  ba  Ua  is  a  vector,  and  the  only  scalar  term  in  the  expres- 
sion is  c. 

9.   It  is  evident  from  Art.  7  that  if  a,  &,  c  are  scalar  coeffi- 
cients, and  tt  any  vector,  we  have 

{a  +  b  +  c)  a=  aa-\-ba  +  Ca   .     .     .     .     (G). 

Furthermore,  if  (Fig.  6) 

OA  =  a,      AB  =  /3,      BC  =  y,      Oa'  =  ma. 


GEOIHETRIC    ADDITION   AND   SUBTKACTION. 
then,  a'b'  being  drawn  parallel  to  ab  and  b'c'  to  BC, 
a'b'  =  7JI/3,     b'c'  =  my. 


Now 


OC  =  a  +  ^  +  y, 


Fig.  6. 


and 


oc'  =  moc  =  m  (a -\-  (3  +  y)  , 
But  we  have  also 


oc'  =  oa'  +  a'b'  +  b'c' 

=  ma  +  m/S  +  my. 


Hence 


7n  (a  +  /3  +  y)  =  t^ioL  +  m/S  +  my      .     .     .     (  7) , 


or  the  distributive  laiv  holds  good  for  the  multiplication  of  scalaY 
and  vector  quantities. 


10.   It  is  clear  that  while 

a  —  a  =  0, 

a  ±  /3  cannot  be  zero,  since  no  amount  of  transference  in  a  direc- 
tion not  parallel  to  a  can  affect  a. 
Hence,  if 

na  +  mft  =  0, 

since  a  and  y8  are  entirel}'  independent  of  each  other,  we  must 
have 

na  =  0     and     mfi  =  0, 
or 

91  =  0     and       m  =  0. 
Or,  if 

ma  +  n(3  =  m'a-j- n' ^, 
then 

m  =  m'     and     n  =  n'. 

And,  in  general,  if 

2a  +  2^  =  0, 

then  [    .     .     .     .     (8). 

2a=0    and    2/3  =  0 


QUATERNIONS. 

Three  or  more  vectors  ma}-,  liowcvcr,  neutralize  eaeh  other. 
Thus    (Fig.  7) 

^'^■''-  a  +  ^  +  y  +  8=0, 

€-/3-a=0, 

ii^^-^O-^           Y       and   this   whether  abcd  be  plane  or 
^/\y^''  gauche.     In  any  closed  figure,  there- 

_^ ,        fore,  we  have 

a  +  /3  +  y  +  S+ =0, 

where  a,  (3,  y,  8,  ,  are  the  vector  sides  m  order. 

11.   Examples. 

1.  The  right  lines  joining  the  extremities  of  equal  and  jKtrallcl 

right  lines  are  equal  and  parallel. 

Fig.  8. 

q ^^^ _^  Let  OA  and   no   (Fig.  8)  be 

the   given   lines,   and   oa  =  a, 
^^     J\^     /^y  BO  =  yS,    DA  =  y.      Then,    b}- 

condition,  ud  =  a. 
Now, 

BA  =  BO  +  OA  =  ^  +  a  ; 

also,  ,  , 

BA  =  BD  +  DA  =  a  -|-  y  ; 

or,  equating  the  values  of  ba, 

/3  +  a  =  a  +  y. 

Hence  (Art.  2),  y  =  ^,  and  bo  is  parallel  and  equal  to  da. 

2.  The  diagonals  of  a  parallelogram  bisect  each  other. 

In  Fig.  8  we  have 

bd  =  oa  =  OP  +  PA ; 

also 

BD  =  BP  +  PD  ; 
.'.    OP  +  PA  =  BP  +  PD. 

But,  OP  and  pd  being  in  the  same  right  line, 

OP  =  ?«PD. 

Similarly 

"  PA  =  »BP. 


GEOMETRIC   ADDITION  AND   SUBTRACTION. 


Hence 


and 


mPD  +  nuF  =  PD  +  BP, 

7ft  =  1 ,     n=  1, 

OP  =  PD,       BP  =  PA. 


3.  If  tioo  triangles,  having  an  angle  in  each  equal  and  the 
inchiding  sides  proportional,  be  joined  at  one  angle  so  as  to  have 
their  homologous  sides  parallel,  the  remaining  sides  will  he  in  a 
straight  line. 


Let  (Fig.  9)  AB  =  a,  AE  =  j3.    Then, 
by  condition,  do  =  xa,  db  =  x^. 
Now 


Fif?.  9. 


CB  =  CD  -|-  DB  =  CC   (/?  —  a)  , 


But 


BE  =  ^  —  a. 


Hence  (Art.  2),  b  being  a  common  point,  cb  and  be  are  one 
and  the  same  right  Hne. 


4.  If  tivo  right  lines  join  the  alternate  extremities  of  two 
parallels,  the  line  joining  their  centers  is  half  the  difference  of 
the  parallels. 


We  have  (Fig.  10) 

AB  =  AD  +  DC  +  CB, 

and,  also, 

AB  =  AE  +  EF  +  FB. 

Addino; 


Fig.  10. 


or,  as  lines. 


2  AB  =  (ad  +  Ae)  +  (dC  +  Ef)  +  (CB  +  Fb) 
=  EF  —  CD  ; 


AB  =  ^  (eF  —  CD)  , 


10 


QUATERinONS. 


5.    The  meclials  of  a  triangle  meet  in  a  point  and  trisect  each 
other. 


^'"■""  Let  (Fig.   11)  BO  =  a,  CD  =  /3.     Then 

OC  =  a,  DA  =  (3. 

Now 

BA=2a+2/3=2(a  +  (3), 

and,  since  od  =  (a  +  /3).  ba  and  od  are 
parallel. 
Again 

BP  +  PA  =  BA  =  2  OB  =  2  (op  +  Pd)  . 

But  BP  and  pd,  as  also  op  and  pa,  lie  in  the  same  direction, 

and  therefore 

BP  =  2  PD     and     pa  =  2  op. 

Hence  the  medials  oa  and  db  trisect  each  other. 
Draw  cp  and  pe.    Then 


and 


BP=2PD  =  fBD  =  |(2a  +  ^), 

CP=CB  +  BP=|  (2a  +  y8)-2a  =  |-  (/3  -  a)  , 
PE  =  PB  +  15E  =  a  +  /?  -  f  (2  a  +  /5)  =  i  (/3  -  a)  , 


Hence  pe  and  cp  are  in  the  same  straight  lino,  or  the  medials 
meet  in  a  point. 

6.   In  any  qnadrilateral,  plane    or  gaitcJie,  the  bisectors  of 
opposite  sides  bisect  each  other. 

"We  will  first  find  a  value  for  op  (Fig.  12)  under  the  supposi- 
tion that  p  is  the  middle  point  of 
ge.  We  shall  then  find  a  value  for 
OP,  under  the  supposition  that  p  is 
the  middle  point  of  fii.  If  these 
expressions  prove  to  be  identical, 
these  middle  points  must  coincide. 
In  this,  as  in  man}'  other  pi'oblems, 
the  solution  depends  upon  reaching 
the  same  point  b}'  difierent  routes  and  comparing  the  results. 


Fisr.  1-2 


GEOMETRIC   ADDITION  AND   SUBTRACTION.  11 

Let  OA  =  a,  OB  =  |S,  OC  =  y. 
1st.  OC  +  CG=OE+EG.  (a) 

But 

which,  in  (a),  gives 

y  +  i(f3  —  y)=ia  +  Ea. 
.-.    EP  =  ^EG  =  i(y  +  ^  — a), 
OP  =  OE  +  EP  =  |a  +  i  (y  +  y8  -  a) 

=  i(a  +  /3  +  7).  (&) 


or 


2d.  FH  — |-AB  =  FO  +  OA, 

rH-|-(/3-a)  =  -|y  +  a. 
.'.    FP  =  |FH  =  :|:(a  +  ^  — y), 
OP  =  OF  +  FP  =  ly  +  i  (a  +  ^  —  y) 

=  i(a  +  /3  +  y), 

which  is  identical  with   (5).     Hence,  the  middle  points  of  fh 
and  GE  coincide. 

7.  If  ABCD  (Eig.  13)  be  any  parallelogram,  and  op  a^iy  line 
parallel  to  dc,  and  the  indicated  lines  he  draiun,  then  will  mn 
be  p^ct^rallel  to  ad. 

Let  AM  =  a,    BM  =  fi. 

Then 

AO  =  ma, 

AD  =  na  +_2)y3, 

OD  =  —  ma  +  7la  +p>l^' 

"We  have 

NM  =  NO  +  OM  =  NP  +  PM, 

in  which 

NO  =x{—  ma  +  ?Za  +  pfi)  , 

OM=  (1  —  m)  a, 

NP  =  aj  (  —  ?«/?  +  ?la  +pjB)  , 

P3I  =  (1  —m)j3. 


12  QUATERNIONS. 

Substituting  in  the  above  equation,  we  obtain,  by  Art.  10, 

1  —  7n 

X  = 

111 

Substituting  this  value  in 

KM  =  NO  -f  CM, 

( —  ma  +  na  +i>/5)  +  (1  —  in)  a 


Fig 

.  13. 
M 

T> 

^ 

C 

A 

\/ 

7\ 

B 

/ 

L 

V^ 

N 

KM  = 

1 

—  m 
1)1 

1_ 

—  m 

m 


{na  +Pp)  = AD. 


m 


Hence  ad  and  nji  are  parallel. 

8.  If,  through  any  jJOint  in  a  2^ciranelog7'am^  lines  be  drawn 
parallel  to  the  sides,  the  diagonals  of  the  two  non-adjacent 
2Kirallelograms  so  formed  will  intersect  on  the  diagonal  of  the 
original  parallelogram . 


Fig.  14. 
A  I' 


and 


Let  (Fig.  14)  OA  =  a,  on  =  /8. 
c      Then  oii  =  ma,  oe  =  ??/?. 
AVc  have 

RD=RO +OK -I-  Er)= ??/34-  ( 1  —m)  a, 
ES  =E0  +  oK  +  us  =ma+  i\  —  n)  /?. 

Also 

FO  =  FR  +  RO  =  CTRD  +  RO  =  o;  [»/3  +  ( 1  —  ??? )  a]  —  ma,     (a) 


FO  =  FE  4-  EO  =  ?/ES  4-  EO  =  ?/  [??la  +  (1  —  7i)  /?]  —  n{3.       (6) 

From  (a)  and  (b) 

nx  =  y  (l  —  n)  — 71     and    x{l—m)  —  m  =  ym. 
Eliminatinj?  y 


X  = 


1  —  ??l  —  M 


GEOMETRIC   ADDITION   AND   SUBTEACTION, 
Substituting  this  value  of  x  in  (a) 

FO  = [n/3  +  (1  —  in)  a]  —  ma 

(/3  +  a), 


13 


1  —  m  —  71 

mn 


1  —  m 
or,  FO  and  oc  =  (^  +  a)  are  in  the  same  straight  line 


9.  7/",  in  any  triangle  oab  (Fig.  15),  a  line  od  he  drawn  to 
the  middle  point  of  as,  and  he  produced  to  any  point,  as  f,  and 
the  sides  of  the  triangle  be  produced  to  meet  af  and  bf  in  h  and 
R,  theii  ivill  HR  be  parallel  to  ab. 


Let  OA  =  a,  OB  =  /?.     Then  or  =  xa, 
OH  =  yj3,  AB  =  (3  —  a. 

Now  o 

OD  =  OA  +  ^  AB  =  |-  (a  +  /5)  . 


Also,  OF  =  z  (a-f^),  that  is,  some 
multiple  of  od. 
Then,  1st. 

BR=pBF, 

—  f3  +  Xa:=p  (  — /?  +  Of) 

=p[-/?  +  ^(^  +  /3)]; 

.-.  x=2)z     and     —l=pz—p. 


Fis.  15. 


Eliminating  z 
And,  2d. 


p  =  cc  +  1. 


AH  =  gAF, 
-a  +  ?//?=g  (-a  +  OF) 

=  g[_a  +  2;(a  +  /3)]; 
.-.  y  —  qz     and     —  1  =  g2  —  g. 


Eliminating  2 
From  (a)  and  (6) 


g  =  2/  +  i. 


cc      ?/ 


(a) 


(&) 


14 


QUATERNIONS. 


and,  since  j)  =  a'  +  1   and  q  =  i/  +  1, 
Fig.  15. 


a;  =  y     and    P  =  Q- 

••  KII  =  KO  +  OH  =  y/3  —  Xa  =  X  (/3  —  a) 

=  .r.vu, 

or,  ini  and  ah  are  parallel. 


10.  If  any  line  pr  (Fig.  IG)  be  draicn,  cutting  the  tiro  sides 
of  any  triangle  abc,  and  he  produced  to  meet  the  third  side  in  q, 
then 

^'S-  16.  PC  .  BQ  .  KA  =  CR  .  AQ  .  liP. 


Let  ijp  =  a,  CR  =  (3.     Then  rc=pa, 
RA  =  rfi  and  ba  =  hc  +  ca  =  (1  +^>)  a 

+  (!+>•)/?• 
"We  have 


AQ  =  .TBA  =  X  [(1  +  ;>)  a  +  (1  +  r)  ft], 

as  also 

AQ  =  AR  +  RQ  =  —  rft  +  ?/I'R  =  —  rft  +  ?/  (;)a  +  ft)  . 

.-.  X  (1  +2))  =  yp  aiifi  ^'  (1  +  '■)  =  -  '■  +  y- 


Eliminating  y 
whence 

or 


a-=  (1  +  x)pr; 

AQ_BQ   PC   RA 
BA   BA   BP   Cr' 

PC  .  BQ  .  RA  =  CR  .  AQ  .  BP. 


11.   If  triangles  are  equiangular,  the  sides  about  the  equal 
angles  are  proportional. 

Let  (Fig.  17)  15C  =  a,   c\  =  ft.      Then   be  =  ?na,   ed  =  nft, 
BD  =  ma  +  nft  and  nx  —  a  +  ft. 
Now 

BD  =  pBA, 

ma  +  nft  =  ])  {a  +  ft). 


"WTience 


m  =  p,     n  =  j>     and     m 
.*.   BE  :  BC  :  :  ED  :  CA. 


GEOMETBIC    ADDITION   AND    SUBTRACTION.  15 

12.    If,  throvgh  any  point  o  (Fig.  17),  xvithin  a  triangle  abc, 
lines  he  draivn  piarallel  to  the  sides,  then  icill 

^  +  ^^  +  ^  =  2. 

CA         CB         AB 

Let  CA  =  /3,  CB  =  a.  Then  ab  = 
a  —  /?,  ED  =  m/3,  HI  =  j)  (a  —  /5)  and 
GF  =  na. 

We  have 

CO  =  CG  +  GO  =  cii  +  HO.  (a) 

Now,  as  Imes, 

^^  <5A  /-.NO 

= =  n,  .'.     CG=:  CA  —  GA=  (1— 70  p. 

CB         CA  V  /  r 


EB  ED 

CB  CA~       ' 

DB  _  DE 

AB  AC             ' 


.'.    GO  =  CE  =  CB  —  EB  =  (1—  m)  a. 

.-.    HO  =  AD  =  AB  —  DB  —  (1  —  m)  (a  —  (3). 

Substituting  in  (a) 

(1  _  ,,)  /3  +  (1  -  on)  a  =  pf3  +  {l-  m)  (a  -  ^) , 
or  (Art.  10)  ,        ,  o 

12.  Complanar  vectors  are  those  which  lie  in,  or  parallel  to, 
the  same  j^htne.  If  a,  ^,  y  are  any  vectors  in  space,  thej^  are 
complanar  when  equal  vectors,  drawn  from  a  common  origin, 
lie  in  the  same  plane. 

If  a,  y8,  y  are  complanar,  but  not  parallel,  a  triangle  can  al- 
ways be  constructed,  having  its  sides  parallel  to  and  some  mul- 
tiple of  a,  /3,  y,  as  aa,  b/3,  cy.  If  we  go  round  the  sides  of  the 
triangle  in  order,  we  Ijave 

aa  +  5^  -f  cy  =  0. 

If  a,  /3,  y  are  not  complanar,  conceive  a  plane  parallel  to 
two  of  them,  as  a  and  (3.  In  this  plane  two  lines  may  be  drawn 
parallel  to  and  some  multiple  of  a  and  f3,  as  aa  and  h/S ;  and 
these  two  vectors  may  be  represented  by  pS  (Art.  3) . 


16  QUATERNIONS. 

Now  p8,  being  in  the  same  plane  with  aa  and  b(3,  cannot 
theivfore  be  equal  to  y,  or  to  any  multiple  of  it ;  j?S  and  y  can- 
not therefore  (Art.  10)  neutralize  each  other.     Hence 

2)8  +  cy  =  aa  +  Z>/3  +  cy        cannot  be  zero. 

If,  then,  ice  have  the  relation 

aa  +  b(S  +  cy  =  0 

hetiveen  non-parallel  vectors,  they  are  complanar;  or,  if  a,  (i,  y 
be  not  complauar,  and  the  above  relation  Ije  true,  then,  also, 

a  =  0,       b  =  0,       c  =  0. 

13.  Co-initial  vectors  are  those  ivhich  denote  transference 
from  the  same  pioint. 

(a).    If  three  co-initial  vectors  are  complanar,  and  give  the 

relations,  .  .  ,   i  o  ,  a  -> 

(a)     aa  +  h(3  +  cy  =  0  \  ,^. 

(6)     a  +  b  +  c  =  0        J ^  ^' 

they  xoill  terminate  in  a  straight  line. 

For,  let  ox  — a  (Fig.  15) ,  ob  =  /3,  od  =  y.     Then  da  =  a  —  y, 

BA  =  a  —  /?. 

From  Equation  (9),  {b) 

(a  +  &  -f  c)  a  =  0, 
from  which,  subti'acting  (a)  of  Equation  (9), 

6  (a  -  /?)  +  C  (a  -  y)  =  0, 

6ba  +  CDA  =  0  ; 

and,  since  these  two  vectors  neutralize  each  other,  and  have  a 
common  point,  thej'  are  on  the  same  straight  hue.  Hence, 
A,  D  and  15  arc  in  the  same  straight  line. 

(6).  Conversely,  if  a,  f3,  y  are  co-initial,  complanar  and  ter- 
minate in  the  same  straight  line,  and  a,  b,  c  have  such  values 
as  to  render  aa -f  6^ -f- cy  =  0, 

^^'^''  ^'"^  a  +  6  +  c  =  0. 

DA  =  a  —  y     and     BA  =  a  —  /?. 


GEOIVIETEIC   ADDITION   AND   SUBTRACTION. 
But,  by  condition, 


17 


or 

in  which 

14.  Examples. 


a  —  (3  =  X  (a  —  y) , 

{1  —  x)  a  —  /3  +  Xy  =  0, 

(1  —  a-)  —  1  +  a;  =  0. 


1.  The  extremities  of  the  adjacent  sides  of  a  parallelogram 
and  the  middle  'point  of  the  diagonal  between  them  lie  in  the  same 
straight  line. 


Fi-  IS. 


Let   OA 
Then 


a,    OB   =   /3,    OC  =  y. 

OD  =  OB  +  BD, 
2y-/3-a  =  0. 


But,  also,  2  —  1  —  1  =  0 

hence,  b,  c  and  a  are  in  the  same  straight  line  (Art.  13). 

2.  If  two  triangles,  abc  and  smn  (Fig.  19),  are  so  situated 
that  lines  joining  corresponding  angles  meet  in  a  point,  as  o, 
then  the  pairs  of  corresponding  sides  x>roduced  will  meet  in  three 
points,  p,  Q,  R,  ivhich  lie  in  the  same  straight  line. 

Let    OA  =  a,    OB  =  /?,    OC  =  y. 
Then     os  =  ma,  om  =  nji, 

ON  =  py,  BA  =  a  —  /?, 

MS  =  ma  —  nfi, 

BR  =  a;  (a  —  fS)  and 


1st. 


BM  =  BR  —  MR, 


or 


n{3  —  I3  =  x  (a  —  (3)  —y  (ma  -  n(3) , 
.'.  7i  —  1  =  —  cc  +  yn,  X  —  my  =  0. 


Eliminating  y 


m  {n  —  V) 


18                                            QUATERNIONS. 

Also 

m  (n  —  1) 

OU  =  OB  +  BR  -  /?  -h  X    {a  -  (i)   -  (i             ^^^  _  ^^         (a 

-/?), 

whence                         „  (^^-1)  (i-m  {n  -\)  a 

OR  = 

7/6   —    IL 

(«) 

2d.                                                  CN  =  CP  —  NP, 

or 

2iy-y=v  {jB-y)-w  {n(3  -  X)y). 
.    p  —  1  =  —  V  +  v:}),         V  —  wn  =  0. 


Eliminating  w  ,(  ( ^,  _  i ) 

V  = 
Also 


n  —  jj 


9?  (p  —  1) 

OP  =  OC  +  CP  =  y  +  r  (/?  -  y)  =  y ^^  _  ^^       (/?  -  y), 

^^^^^^  ^^_j>(»-l)y-.(;>-l)/3  (^^ 

3cl.    In  the  same  manner,  we  obtain 

-//i  (/(  —  1)  a  —  p  (?H  —  1)  y 


OQ  = 


2)  —  til 


(c) 


From  (a) ,  (b)  and  (c)  vrc  observe  that,  clearing  of  fractions, 
and  multiplying  (o)  by  i>  —  1,  ('->)  by  ?/i  —  1,  (c)  by  7i  —  1,  and 
adding  the  three  resulting  equations,  member  by  member,  the 
collected  coefficients  of  a,  (3,  y,  in  the  second  member  of  the 
final  equation,  are  separately'  equal  to  zero.  Hence  the  fu-st 
member 

OR  (m  —  n)  (2)  —  1)  +  OP  (n  —  p)  (m  — 1)  +  oq  (p—m)  (n  — 1)  =  0. 

But 
(m  -  »)  (;)-!)  +  (n  -p)  (m  -  1)  +  (P  -  m)  (?i  -  1)  =  0. 

Hence,  r,  p  and  q  are  in  the  same  straight  line.     ' 


GEOMETEIC    ADDITION   AND    SUBTKACTION. 


19 


3.    Given  the  relation 

aa  -j-  b(3  -\-  Cy  =  0. 

Then  a,  ft,  y  are  eomplanar ;  but,  if  co-initial  (as  they  may 
be  made  to  be,  since  a  vector  is  not  changed  by  motion  parallel 
to   itself,  i.e.  hy  translation 

without  rotation) ,  and  a  +  '^"     ' 

6  +  c  is  not  zero,  they  do 
not  terminate  in  a  straight 
line.  Hence,  if  o  is  the  ori- 
gin, and  A,  B,  c,  their  ter- 
minal points,  A,  B  and  c 
are  not  collinear.  Let  these 
points  be  joined,  forming 
the  triangle  abc  (Fig.  20), 
and  OA,  OB,  oc  prolonged  to 

meet  the  sides  in  a',  b^  c!     To  find  the  relation  between  the 
segments  of  the  sides,  let 


whence 


OA'=a'=ri'a,        ob'=/3'=  ?//?,        oc'=y'=2!y, 

«'  p       /8'  y' 

a=-,  p  =  -,  7=-- 


Substituting  these  in  succession  in  the  given  relation, 


-a'+6^-f-Cy  =  0, 


aa  +  -B'+  Cy  =  0, 

y 


whence,  since  a,'  c,  b  are  to  be  collinear, 


_  +  6  +  c  =  0, 

X 


20  QUATERNIONS. 

and,  for  a  like  reason, 

h 
a-\ 1-  c  =  0, 

y 

a  +  i  +  -  =  0. 
z 

Whence 

«  „  ^  -,.  p 

h  +  c  a  +  c  a  +  b 

and 


h-Y-c  a  +  c  a  +  h 

or,  from  the  given  relation, 

,_  h(i  +  Cy  „,_  Cy  4-  Ct«  ,_  «a  +  hji 

°-~    h  +  c  '         ^~    c  +  a  '         ■^"~    a  +  6  ' 

"Whence 

6   (a'-^)=C  (y-a'), 

C    (^'-y)=a(a-^'), 

a(y'-a)=H/3-7'), 

and 

ba'  _  c         cb'  _  a         Ac'  _  6 
a'c  ""  V       b'a  ~~  c'        c'b       a' 

or,  multiplying, 

ba'  .  cb'  .  Ac' .  =  a'c  .  b'a  .  c'b. 

4.  If  o  (Fig.  20)  he  any  pointy  and  abc  any  triangle,  the 
transversals  through  o  and  the  vertices  divide  the  sides  into  seg- 
ments having  the  relation 

ba'  .  cb'  .  ac'  .  =  a'c  .  b'a  .  c'b. 

Let  a'c  =  a,  bc  =  aa,  cb'=  (3,  ca  =  h(3.     Then  ba  =  aa  +  h/S. 

Also  let 

BO  =  XBbJ  OA  =  ^a'a,  BC'=  ?JIBA,  CC'=  ZCO. 


GEOMETRIC   ADDITION   AND   STJBTEACTION. 


21 


Then 

BO  =  xbb'  =  X  (bc  +  cb')  =  X  (aa  +/S)  , 

OA  =  2/a'a  =  y  (a'c  +  ca)  =?/(a  +  6/5), 

Bc'=  mBA  =  m  (aa  +  b/3) , 

cc'=  zco   =  z  (cB  -f-  bo)  =  z  [—  aa  +  x  (aa  +  /5)] . 

From  the  triangle  boa  we  have 

bo  +  OA  +  AB  =  0, 

X  (aa -\- f3)  +  7j  (a  +  bfS)  —  bj3  —  aa  =  0. 
.'.     xa-\-y  —  a=0,         x  -\-yb  —  b  =  0. 


Eliminating  y 

From  the  triang-le  bcc' 


6(1- a) 
1  —  ba 


bc  +  cc'+  c'b  =  0, 
aa-i-z  [—  aa-{-x  (aa  +  ;8)]  —  m  (aa  +  b(3)  =  0, 

whence,  as  usual,  and  substituting  the  above  value  of  x, 

.  6(1— a)  1  — a 

l  —  ')n  =  z  —  z—^ — ; — -,       m  =  z- 


or 


l-6a  ' 
1  —  m      1—6 


1—  6a 


m         1—  a 
Substituting  for  m,  6  and  a, 


C  A_  AB'        CA' 

Bc'"~  b'c      a'b' 
which  is  the  required  relation. 


5.  If  (Fig.  20)  U7ies  be 
draion  through  a'  •&',  c'  and 
produced  to  meet  the  opposite 
sides  of  the  triangle  in  f,  q, 
R,  then  are  p,  q  and  r  col- 
linear. 


Fig.  20. 


22  QUATERNIONS. 

"With  the  notation  of  the  last  example, 

Bc'  =  mux  =  — — — -—  (cia  +  bB) . 
a-\-b  —  2 

1st.    From  the  triangle  c'ba' 


c'a'=  c'b  +  B\' 


a  +  b 
a  +  b -2 


-(aa  +  &^) +  («-!)  a 


~l(b-2)a-b(3l 


Also 


a'r  =  a;c'A'=  a'c  +  cr  =  a'c  —  yf3, 
a -I 
+ 


^    1;       Aih-2)a-bf3:\  =  a-yf3, 
a  +  b  — 2 


b 

y  = 


b-2' 

'''"'^  BR  =  BC  +  CR  =  aa  -  -A_  R,  (a) 

b-2'^  ^  ^ 

2cl.    From  the  triangle  c'ab' 

c'b'=  c'a  +  ab' 

=  (l-7n)(aa  +  6i3)  +  (l-5)/3 
b-\ 


Also 


[«a-(«-2)/3]. 
a  +  b  —  2 

b'q  =  xc'b'=  b'c  +  cq  =  b'c  +  2/a, 

x-^-^=l-{_aa-{a-2)(i-]  =  -li  +  ya, 
a  +  b  — 2 

a 
.-.  y  = 


a-2' 


and 


3d. 


BQ  =  BC  +  CQ  =  (a  +y)a  =  — i —^a.  {0) 

Ct  "~~  ^ 

a'p  =  .rA'B'=  x(a  +  f3), 
a'p  =  a'b  +  v.r  =  {l  —  a)a  +  y{aa  +  bfi) , 
n-1 
a  —  b 


GEOMETRIC    ADDITION   AND    SUBTRACTION.  23 


and 


BP  =  ?/BA  = (aa  +  bB) , 

a  —  b 


(c) 


Multipl^-ing  the  second  members  of  (a),  (h),  (c),  by  (a—l) 
(&  —  2),  —  (a  —  2)  (6  —  1),  (a  —  b)  respective!}',  theii-  sum  is 
zero.     Hence 

(a  - 1)  (5  -  2)  BR  -  (a  -  2)  (&  - 1)  bq  +  (a  -  6)  bp  =  0. 

But 

(a  - 1)  (6  -  2)  -  (a  -  2)  (6  -1)  +  (a  -  &)  =  0. 

Hence  r,  q  and  p  are  collinear. 


6.  If  FC  (Fig.  20)  and  po  be  produced  to  meet  aa'  and  bc, 
tJien  T  CDirt  s  are  collinear  ivith  cl  A  similar  proposition  would 
obtain  for  q  and  r. 

With  the  following  notation, 


we  have 


BA  =  a,         ba'=  /?,         bb'=  aa  +  b^f 


BO  =  BA  +  Ab'+  b'o  =  Ba'+  a'o, 

a  +  6/3  -  (1  -  a)  a  +  x{aa  +  b,8)  =  /3  +  y (a  -  /3). 

a 


■••  y 


^^Jyp  +  aa 


also 


a  +  b    ' 


Fig.  20. 


BP  =  ba'+  a'p  =  BA  +  ap  ; 
/3  +  2[aa+(/j  — l)^]=a  +  Wa, 
a  —  \  +  b 
1-6    ' 
a  a 


and 


1-6' 


BC  =  BA'+  A'C  =  BA  +  AC, 

/3  +  ^J/3  =  a  +  16  [(1  -  a)  a  -  6/3], 


24 


QUATERNIONS, 


I- a 


BC  = 


\-a 


Now  to  find  BS,  Bc'  and  bt,  we  have 


Fig.  20. 


1st. 

BS  =  x'ba'  =  i;r  +  y'vo, 
.     ..,  h 


1 

-26- 

1 

-  a 

BS 

bft 

1- 

-■2b- 

a 

2d. 

BC' 

•. 

v'ba  = 
v'— 

■■  BC  4- 
a 

«co, 

2  a 

+  b- 

1' 

V.C 

,/ 

aa 

2a-\-b-l 


3d. 


BT  =  BA'+  a't  =  Ba'-|-  z'a'o  =  BP  -f-  iv'pc, 


I     n  +  b 


a  —  b 

bB  -  aa 
bt  =  -t! 

b  —  a 
Clearing  of  fractious  and  adding 

(1  -  2  6  -  o)  BS  +  (2  o  -I-  &  - 1 )  Bc'+  (6  -  a)  bt  =  0, 
(\-2b-a)  +  (2a-\-b-l)  +  (h  -a)  =  0. 
Hence  s,  c'  and  t  are  collinear. 


as  also 


15,  A  medial  vector  is  one  drawn  from  the  origin  of  two  co- 
initial  vectors  to  the  middle  point  of  the  line  joining  their 
extremities. 


GEOMETRIC    ADDITION   AXD    SUBTRACTION. 


'ZO 


Thus  (Fig.  21),  if  p  is  the  middle  point  of  ab,  op  is  a  medial 
vector.     To  find  an  expression  for  it,  let  oa  =  a,  ob  =  yS,  then 


or,  adding, 


OP  =  OA  +  AP  =  a  -f-  AP, 
OP  =  OB  +  BP  =/3  —  AP, 


0P  = 


a  +  /3 


(10), 


The  signs  in  this  expression  will,  of  coijrse,  depend  upon  the 
original  assumptions.     Thus,  if  ao  =  a, 

OP  =  —  a  +  AP  =  /3  —  AP, 

op  =  ' 

2 

16.   An  Angle-Bisector  is  a  line  which  bisects  an  angle. 

To  find  an  expression  for  an  angle-bi- 
sector as  a  vector,  let  oe  =  a  (Fig.  21) 
and  of  =  ^  be  unit  vectors  along  oa  and 
OB.  Complete  the  rhombus  oedf.  Since 
tlie  diagonal  of  a  rhombus  bisects  the 
angle,  od  is  a  multiple  of  op.  Now  od 
=  a  +  (3,  hence 


OP  =  x(a-\-(3) 


(11). 


In  this  expression  op  is  of  an}'  length  and  x  is  indeterminate. 
If  op  is  limited,  as  by  the  line  ab,  then 


AP  =  x{a  +  13)  —  aa, 
AP  =  l/AB  =  ?/  (&^  —  aa)  , 

x{a  +  /3)  —  aa  =  y{b/3  —  aa) , 


(a) 


or 


Eliminating  x 


X  —  a  =  —  ya     and     x  =  yb. 
a 

y  =  - 

a 
Substituting  in  (a) 


a  +  b 


a-\-b 


(12). 


26 


QUATERNIONS. 


17.   Examples. 

1.  If  2'>o,'rcdMograms,  whose  sides  are  2'>ciTallel  to  tico  given 
lines,  be  described  upon  each  of  the  sides  of  a  triangle  as  diago- 
nals, the  other  diagonals  icill  intersect  in  a  j^oint. 


Fitr.  22. 


Let  ABC  (Fig.  22)  be  the  given  tri- 
angle. Let  the  diagonals  u'f  and  a'd 
intersect  in  p,  and  suppose  oe  to  meet 
a'd  in  some  point  as  p! 

Let  OA  =  a,  ob'=  /?,  whence  oa'= 
?na,  OB  =  n^. 

Now 

b'p  —  DP  =  a. 

But 


And 


d'p  =  ?/b'q  =  y  •  i  (b'c  +  b'b) 

=  ^y[ma  +  (n-l)/3]. 

DP=  ZDll  =  Z  .^  (dC  +  C.v') 

=  i/[(m-l)a-^]. 


(«) 

(Art,  15) 


Substituting  in  (a) ,  we  obtain,  as  usual, 


Again 
But 


2  (\-n) 

z— — i '- — 

\-\-')>in  —  n 

op'—  dp'=  a  +  (3. 

0P'=  .TOG  =  X  .^  (OA  +  Ob) 

=  ^.r  (a  +  »/3). 


(h) 


Substituting  in  (h)  this  value  of  op'  and  dp'=  i'DII,  we  obtain 

as  before, 

2  (1—  u) 


v  = 


1  +  mti 


Or,  ron  =  zdii  =  dp'=  dp.     Hence,  p  and  p'  coincide,   and 
the  three  diagonals  meet  in  a  point. 

2.    A  triangle  can  alwaj/s  be  constructed  lohose  sides  are  equal 
and  parallel  to  the  medials  of  any  triangle. 


GEOMETEIC   ADDITION"   AND    SUBTRACTION. 


27 


In  Fig.  23  we  have 

aa'=  ab  +  ba'=  ab  +  I^BC. 
bb'=  bc  +  ^CA. 
CC'=  CA  +iAB. 

.*.  aa'+ bb'+cc'=  f  (ab  + BC  + ca)  =  0.   (Art. 
■  3.    The  angle-bisectors  of  a  triangle  meet  in  a  point. 

Let  a,  yff,  y  be  unit  vectors  along  bc,  j,.    23 

AC,  AB  (Fig.  23). 
Then  (Art.  16) 

AP  =  X  (y  +  ^)  , 
BP  =  ?/  (a  -  y)  . 


10), 


Now 


(a) 


BC  =  AC  —  AB, 

aa  =hfi  —  Cy 


(&) 


where  a,  6,  c  are  the  lengths  of  the  sides. 
Substituting  a  from  (6)  in  (a) 


'h(i  -  cy 


We  have  also 


cp  =  AP  —  AC  =  a;  (y  +  ^)  —  b(3, 
fb(S  -cy       \ 

CP  =   BP  +  CB  =  ?/     (    ~ y    1   -I-  cy  _   /j^. 


Eliminating  y 
Substituting  in  (c) 

CP  = 


c6 


a  +  &  +  c 


cb 


a  +  b  -\-c 
_         5 

a  +  ?>  +  c 
^         6 

a  +  6  4-  c 
=  i^(a  +  /S) 


(y  +  ^)-6/? 
[cy_(a  +  6)/3] 
(  —  aa  —  ttyS) 


(c) 


Hence  (Art.  16)  cp  is  an  angle-bisector. 


28 


QUATERNIONS. 


18.  The  Mean  Point  of  any  jwlygon  is  that  to  ichich  the 
vector  is  the  mean  of  the  vectors  to  the  angles. 

Hence,  to  find  the  mean  point,  add  the  vectors  to  the  angles 
and  divide  l\y  the  number  of  the  angles.  Thus,  if  oj,  a,.  03  .... 
be  the  vectors  to  the  angles,  the  vector  to  the  mean  point  is 


a,  4-  ao+  03  +  ....  +a„ 


(13), 


where  n  is  the  number  of  the  angles. 

The  mean  i)oint  of  a  polyedron  is  similarl}-  defined.  It  co- 
incides in  either  case,  as  will  ajipoar  later,  with  the  center  of 
gravity  of  a  system  of  equal  particles  situated  at  the  vertices 
of  the  polygon  or  polyedron. 

19.   Examples. 

1 .  The  mean  point  of  a  tetraedron  is  the  mean  point  of  the 
tetraedron  formed  by  joining  the  mean  points  of  the  faces. 

Let  (Fig.  24)  oa  =  a,  ob  =  /3,  oc  = 
y.  The  vcctore  from  o  to  the  mean 
points  of  the  faces  are 


Ha  +  iS  +  y), 

H^^  +  y). 

hiy  +  P). 

and  that  to  the  mean  point  of  the  tetraedron  formed  b}-  joining 
them  is 


'a  +  ft  +  y      a  +  (S      a  +  y      7  +  ft' 
3         "^     3     "^     3     "^     3 


H«  +  /3  +  y), 


which  is  the  vector  to  the  mean  point  of  gabc. 

The  same  is  true  of  the  tetraedron  formed  by  joining  the  mean 
pouits  of  the  edges  ab,  bo  and  ca  with  o,  since 


'a+/3       13  +  y       g  +  y" 
2  2  2 


=  H«  +  ^  +  7)- 


GEOMETRIC   ADDITION   AND    SUBTRACTION. 


29 


The  above  is,  of  course,  independent  of  the  origin,  and  would 
be  true  were  o  not  talien  at  one  of  the  vertices. 

2.    The  intersection  of  the  bisectors  of  the  sides  of  a  quadri- 
lateral is  the  mean  point. 


Let  (Fig.  25)  oa  =  a,  or  =  /?,  oc  =  y, 
0D  =  8,  OR  =  p.     Then  (Art.  15)  o 

p  =  1  (of  +  oe) 

=  i[i('x  +  S)+Hy  +  /5)] 

=  i(a  +  /8+y  +  3). 


If  o  is  at  A,  then  oa  =  a  =  0,  and 

P  =  H/?  +  7  +  S). 

3.  If  the  sides  (in  order)  of  a  quadrilateral  be  divided  propor- 
tionately, and  a  neiv  quadrilatercd  formed  by  joining  the  points 
of  division,  then  will  both  quadrilaterals  have  the  same  mean 
point. 

Let  a,  /?,  y,  8  be  the  vectors  to  the  vertices  of  the  given 
quadrilatex'al,  from  any  initial  point  o. 

Then,  for  the  vector  to  the  mean  point,  we  have 

i(a  +  /3  +  y  +  8). 

If  m  be  the  given  ratio,  and  a',  ^',  y'  8'  the  vectors  to  the  ver- 
tices of  the  second  quadrilateral,  then 

a'=  a-j-m  (/3  —  a)  =  (l—  m)  a  +  m/3, 

/3'=(l-m)/?  +  my, 

y'=  (1  —  m)y  -\-  mS, 

8'  =  a  +  (1-771)  (8  -  a)  =  8  -  m  (8  -  a)  ; 


whence 


i  (/8'+  y'+  S'-f  a')  =  Ha  4-  /5  +  y  4-  8). 


30 


QUATERNIONS. 


4.    In  any  quadrilateral^  plane  or  gauche,  the  middle  point 
of  the  bisector  of  the  diagonals  is  the  mean  point. 

Let  (Fig.  26)  oa  =  a,  ob  =  /9,  oc  =  y,  os  =  ^y. 
^'^•2«-  Then  (Art.  15) 


OP  =  \  (OQ  +  os) 


5.  If  the  tico  opposite  sides  of  a  quadrilateral  be  divided  pro- 
portionately, and  the  points  of  division  joined,  the  mean  i)oints 
of  the  three  quadrilaterals  ivill  lie  in  the  same  straight  line. 

Let  c'  a'  (Fig.  27)  be  the  points 
of  division,  and  m  the  given  ratio- 
Then,  if  OA  =  a,  HC  =  y,  Oa'=  Wla, 
c'c  =  my,  AB  =  /3  and  o  is  the  in- 
itial point,  the  vectors  to  the  mean 
points  p,  p'  p"  are 

OP   =i(3a  +  2^  +  y), 
OP'  =^[(r,i  +  2)a  +  2/3  +  (2-m)y], 
op"=i[(m  +  3)a  +  2/3+(l-m)y]; 
,        I  —  m  .  s 

•••  pp  =  — r—(7-«)^ 

4 

p  p  =  4  (y  - ")  • 

Therefore,  pJ  p','  p  are  in  the  same  straight  line. 


20.   Exercises. 

1 .  The  diagonals  of  a  parallelopiped  bisect  each  other. 

2.  In  Fig.  58,  show  that  bg  and  cii  are  parallel. 

3.  If  the  adjacent  sides  of  a  quadrilateral  be  divided  propotx 
tionately,  the  line  joining  the  points  of  division  is  parallel  to  the 
diagonal  joining  their  extremities. 


GEOMETRIC   ADDITION   AND   SUBTRACTION.  31 

4.  The  medial  to  the  base  of  an  isosceles  triangle  is  an  angle- 
bisector. 

5.  In  any  right-angled  triangle  abc  (Fig.  58),  the  lines  bk, 
CF,  AL  meet  in  a  point. 

'    6.    Any  angle-bisector  of  a  triangle  divides  the  opposite  side 
into  segments  proportional  to  the  other  two  sides. 
^     7.    The  line  joining  the  middle  point  of  the  side  of  any  paral- 
lelogram with  one  of  its  opposite  angles,  and  the  diagonal  which 
it  intersects,  trisect  each  other. 

-  8.  If  the  middle  points  of  the  sides  of  any  quadrilateral  be 
joined  in  succession,  the  resulting  figiu'e  wUl  be  a  parallelogram 
with  the  same  mean  point. 

9.  The  intersections  of  the  bisectors  of  the  exterior  angles 
of  any  triangle  with  the  opposite  sides  are  in  the  same  straight 
line. 

10.  If  AB  be  the  common  base  of  two  triangles  whose  vertices 
are  c  and  d,  and  lines  be  drawn  from  any  point  e  of  the  base 
parallel  to  ad  and  AC  intersecting  bd  and  bc  iu  f  and  g,  then  is 
FG  parallel  to  dg. 


CHAPTER   II. 

Multiplication   and   Division  of  Vectors,  or  Geometric   Multipli- 
cation and  Division. 

21.   Elements  of  a   Quaternion. 
The  quotient  of  tico  vectors  is  called  (i  Quaternion. 
"We  arc  now  to  see  -what  is  meant  l)y  the  quotient  of  two 
vectors,  and  what  are  its  elements. 

Let  a  and  /?'  (Fig.  28)  be  two  vec- 
tors drawn  from  o  and  o'  respectively 
and  not  lying  in  the  same  plane  ;  and 
let  their  quotient  be  designated  in  the 

usual  wa}'  b^'  — ^. 

\  "Whatever  their  relative  positions,  we 

o''  T      ^''      m^*^}'  alwaj'S  conceive  that  one  of  these 

vectors,  as  /3',  ma^'  be  moved  parallel 
to  itself  so  that  the  point  o'  shall  move  over  the  line  o'o  to  o. 
The  vectors  will  then  lie  in  the  same  plane.  Since  neither  the 
length  or  du'ection  of  /3'  has  been  changed  during  this  parallel 
motion,  we  have  (^  =  /3',  and  the  quotient  of  an}'  two  vectors,  a, 
P',  will  be  the  same  as  that  of  two  equal  co-initial  vectors,  as  a 
and  B.     "We  are  then  to  determine  the  ratio  — ,  in  which  a  and  yS 

lie  in  the  same  plane  and  have  a  conomon  origin  o. 

"V^Hiatever  the  nature  of  this  quotient,  we  are  to  regard  it  as 
some  factor  which  operating  on  the  divisor  produces  the  dividend^ 
i.e.  causes  ji  to  coincide  with  a  in  direction  and  length,  so  that 
if  this  quotient  be  q,  we  shall  have,  by  definition, 

gB  =  a    when    —=q (1^)' 

32  ^^  /? 


GEOMETRIC    MULTIPLICATION   AND   DIVISION.  33 

If  at  the  poiut  o'  we  suppose  a  vector  o'c  =  y  to  be  drawn, 
not  parallel  to  the  plane  aob,  and  that  this  vector  be  moved  as 
before,  so  that  o'  falls  at  o,  the  plane  which,  after  this  motion, 
y  will  determine  with  a,  will  differ  from  the  plane  of  a  and  jj,  so 
that  if  the  quotient 

q  and  q'  will  differ  because  their  planes  differ.  Hence  we  con- 
clude that  the  quotients  q  and  q'  cannot  be  the  same  if  a,  ^  and 
y  are  not  parallel  to  one  plane,  and  therefore  that  the  position 
of  the  plane  of  a  and  yS  must  enter  into  our  conception  of  the 
quotient  q. 

Again,  if  y  be  a  vector  o'c,  parallel  to  the  plane  aob,  but 
differing  as  a  vector  from  (i',  then  when  moved,  as  before,  into 
the  plane  aob,  it  will  make  with  a  an  angle  other  than  boa. 
Hence  the  angle  between  a  and  y8  must  also  enter  into  our  con- 
ception of  q.  This  is  not  onl}-  true  as  regards  the  magnitude  of 
the  angle,  but  also  its  direction.  If,  for  example,  y  have  such  a 
direction  that,  when  moved  into  the  plane  aob,  it  lies  on  the 
other  side  of  a,  so  that  aoc  on  the  left  of  a  is  equal  to  aob,  then 

the  quotient  g'  of  -,  in  operating  on  y  to  produce  a  must  turn  y . 

y 

in  a  du'ection  opposite  to  that  in  which  q=-  turns  /3  to  produce 

a.  Therefore  q  and  q'  will  differ  unless  the  angles  between  the 
vector  dividend  and  divisor  are  in  each  the  same,  both  as  regards 
magnitude  and  direction  of  rotation.  Of  the  two  angles  through 
which  one  vector  may  be  turned  so  as  to  coincide  with  the  other 
is  meant  the  lesser,  and  it  will  therefore,  generally,  be  <  180? 

Finally,  if  the  lengths  of  ^  and  y  differ,  then  -  =  q  will  still 
differ  from  -  =  g!     Therefore  the  ratio  of  the  lengths  of  the  vec- 

y 

tors  must  also  enter  into  the  conception  of  q. 

We  have  thus  found  the  quotient  g,  regarded  as  an  operator 
which  changes  (3  into  a,  to  depend  upon,  the  plane  of  the  vectors, 
the  angle  between  them  and  the  ratio  of  their  len2;ths.     Since 


34  QUATERNIONS. 

two  angles  are  requisite  to  fix  a  plane,  it  is  evident  that  q 
depends  upon  four  elements,  and  performs  two  distinct  opera- 
tions : 

1st.  A  stretching  (or  shortening)  of  /3,  so  as  to  make  it  of 
the  same  length  as  a  ; 

2d.  A  turning  of  /3,  so  as  to  cause  it  to  coincide  with  a  in 
direction, 

the  order  of  these  two  operations  being  a  matter  of  indiffer- 
ence. 

Of  the  four  elements,  tlio  turning  operation  depends  upon 
three  ;  two  angles  to  fix  the  plane  of  rotation,  and  one  angle  to 
fix   the   amount   of  rotation    in   that 
Fig.  28.  plane.      The  stretcliing  operation  de- 

pends onl3'  upon  the  remaining  one, 
I.e.,  upon  the  ratio  of  the  vector 
lengths.      As    depending    upon    four 

'T *"        "  elements  we  observe  one  reason  for 

\  calling  q  a  (juaternion.     The  two  ope- 


,, " "^1 — 15'      rations  of  which  q  is  the  symbol  being 

entirely  independent  of  each  other,  a 
quaternion  is  a  complex  qnantiti/,  decomposable,  as  will  be 
seen,  into  two  factors,  one  of  which  stretches  or  shortens  the 
vector  divisor  so  that  its  length  shall  equal  that  of  the  vector 
dividend,  and  is  a  signless  number  called  the  Tensor  of  the 
quaternion  ;  the  other  turns  the  vector  divisor  so  that  it  shall 
coincide  with  the  vector  dividend,  and  is  therefore  called  the 
Versor  of  the  quaternion.  These  factors  are  symbolicalh'  repre- 
sented bj-  Tq  and  Vq,  read  "  tensor  of  q"  and  "versor  of  q" 

and  q  ma}'  be  written 

q  =  Tq  .  Tq. 

22.  An  equality  bettoeen  tico  quaternions  may  be  defined  di- 
rectly from  the  foregoing  considerations. 

If  the  plane  of  a  and  (3  be  moA-ed  parallel  to  itself;  or  if  the 
angle  aob  (Fig.  "28),  remaining  constant  in  magnitude  and  esti- 
mated in  the  same  direction,  be  rotated  about  an  axis  tlu'ough  o 
perpendicular  to  the  plane  ;  or  the  absolute  lengths  of  a  and  y3 


GEOMETKIC   ISrULTIPLICATION   AND   DmSION.  35 

vaiy  so  that  their  ratio  remains  constant,  q  will  remain  the  same. 
Hence  if  , 

a  T       a'  I 

-  =  g     and     — -=  q, 
then  will 

when 

1st.  Tlie  vector  lengths  are  in  the  same  ratio,  and 

2d.  The  vectors  are  in  the  same  or  parallel  planes,  and 

3d.    The  vectors  make  with  each  other  the  same  angle  both  as 

to  magnitude  and  direction. 

The  plane  of  the  v,ectors  and  the  angle  between  them  are 

called,  respectively,  the  plane  and  angle  of  the  quaternion,  and 

the  expression  -,  a  geometric  fraction  or  quotient.     It  is  to  be 

observed  that  q  has  been  regarded  as  the  opei'ator  on  /?,  produc- 
ing a.  This  must  be  constantl}'  borne  in  mind,  for  it  will  sub- 
sequentl}'  appear  that  if  we  write  q/3  =  a  to  express  the  operation 
b}'  which  q  converts  /3  into  a,  q(3  and  I3q  will  not  in  general  be 
equal. 

23.  Since  q,  in  operating  upon  /5  to  produce  a,  must  not  only 
turn  /3  through  a  definite  angle  but  also  in  a  definite  direction, 
some  convention  defining  positive  and  negative  rotation  with 
reference  to  an  axis  is  necessary. 

B}'  x)ositive  rotation  with  reference  to  an  axis  is  meant  left- 
handed  rotation  when  the  direction  of  the  axis  is  from  the  plane 
of  rotation  towards  the  eye  of  a  person  who  stands  on  the  axis 
facing  the  plane  of  rotation. 

[If  the  direction  of  the  axis  is  regarded  as  from  the  eye 
towards  the  plane  of  rotation,  positive  rotation  is  righthanded. 
Thus,  in  facing  the  dial  of  a  watch,  the  motion  of  the  hands  is 
positive  rotation  relatively  to  an  axis  from  the  q\q  towards  the 
dial.  For  an  axis  pointing  from  the  dial  to  the  e^-e,  the  motion 
of  the  hands  is  negative  rotation.  Or  again,  the  rotation  of  the 
earth  from  west  to  east  is  negative  relative  to  an  axis  from  north 
to  south,  but  positive  relative  to  an  axis  from  south  to  north.] 

On  the  aljove  assumption,  if  a  person  stand  on  the  axis,  fac- 
ing the  positive  direction  of  rotation,  the  positive  direction  of 


36 


QUATERNIONS. 


Fig.  31  (bis). 
J 


the  axis  will  always  be  from  the  place  where  he  stands  towards 

the  left. 

K  ?,  A-,  j  (Fig.  31)  be  three  axes  at  right  angles  to  each  other, 
with  directions  as  indicated  in  the 
figure,  then  positive  rotation  is  from  i 
to  ^*,  from  j  to  k,  and  from  k  to  i,  rela- 
tiveh'  to  the  axes  A',  ?',  j  respcctiveh'. 
A  prcciseh"  opposite  assumption  would 
be  equally  proper.  The  above  is  in 
accordance  with  the  usual  method  of 
estimating  positive  angles  in  Trigo- 
nometry and  Mechanics. 

24.   Let  OA  and  on  (Fig.  29)  be  any 

two  co-initial  vectors  whose  lengths  are  a  and  6,  a  and  /?  being 

unit  vectors  along  oa  and  oa,  so  that 

OA  =  aa, 

OB  =  &/3. 

Let  the  angle  boa   between   the 
vectors  be  represented  by  ^ ;   also 
draw  AT)  perpendicular  to  ob,  and 
let  the  unit  vector  along  da  be  8. 
/£  ~  The     tensor    of    on    is    evidently 

a  cos  (^  and  that  of  da  a  sin  <f>.  If 
we  assume  that,  as  in  Algebra,  geometrical  quotients  which 
have  a  common  divisor  are  added  and  subtracted  by  adding  and 
subtracting  the  numerators  over  the  common  denominator,  so 
that 


then,  since 
we  have 


OA 
OB 


OA  =  OD  +  DA, 
CD  +  DA 


OB 
a  cos  </>  .  /? 

a  /'cos  <^ 
h 


_  OD        DA 
~  OB        OB 


+ 


P 


+ 


a  sin  <^  .  S 

sin  <i>  .  S\ 


^ 


GEOMETRIC   MULTIPLICATION    AND   DIVISION.  37 

We  liaA'e  already  defined  (Art.  8)  the  quotient  of  two  parallel 
vectors  as  a  scalar,  and  in  the  first  term  of  the  parenthesis,  jB 

beino;  a  unit  vector,  "  =  1 ,  and 

ox      a  f       ,   ,     .     ,      8\  .  . 

—  =  7   cos<^  +  sm</)  .  -  .  (a) 

OB      b\  pj 

The  last  term  contains  the  quotient  -  of  two  unit  vectors  at 

right  angles  to  each  other.  This  quotient  is  to  loe  regarded,  as 
before,  as  a  factor  which,  operating  on  the  divisor  /?,  produces 
S,  I.e.,  tm'ns  /3  left-handed  through  an  angle  of  90°  ;  and  this 
quotient  must  designate  the  plane  of  rotation  and  the  direction 
of  rotation.  If  we  define  the  eflTect  of  an}'  unit  vector,  operating 
as  a  multiplier  upon  another  at  right  angles  to  it,  to  be  the  turn- 
ing of  the  latter  in  a  positive  direction  through  an  angle  of  90° 
in  a  plane  perpendicular  to  the  operator,  then  the  unit  vector  €, 
drawn  from  o  peipendicular  to  the  plane  of  8  and  yS,  and  in  the 
direction  indicated  in  the  figure,  will  be  the  factor  which  oper- 
ating on  /?  produces  S,  and 

e/3  =  8     or     -  =  e. 

The  unit  vector  e,  as  an  axis,  determines  the  plane  of  rotation  ; 
its  direction  detennines  the  direction  of  rotation,  and  b}'  defini- 
tion its  rotating  effect  extends  through  an  angle  of  90°  ;  as  a 
quotient,  therefore,  it  completely  determines  the  operator  which 
changes  /?  into  8.     Equation  (o)  thus  becomes 

—  =  -  (cos  ^  +  €  sin  </)) , 

OB        h 

or,  if  OA  and  ob  be  themselves  denoted  by  a  and  /?,  and  the  ten- 
sors of  a  and  /3  by  Ta  and  T^S, 

?  =  ^  (cos  <^ -f  e  sin  <^)      ....     (15), 


45l)4;j 


38  QUATERNIONS. 

To 
in  which   —  is  the  tensor  of  g,  being  the  ratio  of  the  vector 

lengths,  and  cos  <^  +  €  sin  <f>  is  the  versor  of  q,  its  plane,  deter- 
mined b}-  the  axis  e,  and  angle  <{>  being  the  plane  and  angle  of 
the  quaternion. 

"When  a  and  (3  are  of  the  same  length,  or  Ta=T/?,  Tq=  —  =  1, 

and  the  effect  of  g  as  a  factor,  or  operator,  is  simply  one  of 
version. 

Like  T,  the  s^-mbol  TJ  is  one  of  operation,  indicating  the  oper- 
ation of  taking  the  versor,  so  that 

Ug  =  cos  </)  +  e  sin  (j>. 

This  operation  takes  into  account  but  one  of  the  two  distinct 
acts  which  we  have  seen  the  quotient  q  must  perform,  as  an 
agent  converting  /8  into  a,  nameh*,  the  act  of  version ;  it  thus 
eliminates  the  quantitative  element  of  length.     In  this  respect  it 

is  similar  to  the  reduction  of  a  a'cc- 
tor  to  its  unit  of  length,  an  oi)cra- 
tion  which  also  eliminates  this  same 
element  of  length,  and  has  been 
designated  by  the  same  sj-mbol  U. 
"When  a  and  /3  are  at  right  angles 
/  to  each  other,  <i  =  90°  and  tlie  ver- 

sor  cos  ^  4-  e  sin  <^  reduces  to  the 
unit  vector  c,  which  has  been  de- 
fined, as  an  operator,  to  be  a  versor  turning  a  line  at  right 
angles  to  it  tln-ough  an  angle  90?  Am'  vector,  therefore,  as  a, 
contains,  in  its  unit  A^ector  in  the  same  direction,  a  versor 
element  or  factor  of  which  Ua  is  the  symbol,  U  indicating  the 
reduction  of  a  to  its  unit  of  length  or  the  taking  of  its  versor 
factor.     Hence  the  appellation  versor  of  a  (Art.  7). 

If  in  Equation  (15)  the  vectors  be  reduced  to  the  unit  of 
length, 

U/3  13 


GEOarETEIC   IMXTLTIPLICATIOX   AI^TD  DIVISION,  39 

25.   AVe  may  now  express  the  relation 

I  =  ^  (cos  <^  +  e  sin  </>)  =  g  (Eq.  15) 


in  the  symloolic  notation 

P        /3       ^l (16), 


or 

q  =  Tq,  Vq 

and  say  that  the  quotient  of  two  vectors  is  the  product  of  a  tensor 
and  a  versor;  and  that 

1st.  The  tensor  of  the  quotient,  (  — ),  is  the  ratio  of  their 
tensors;  ^  ^-^ 

2d.  The  versor  of  the  quotient,  (cos  ^  +  e  sin  ^) ,  is  the  cosine 
of  the  contained  angle  2')lus  the  2Woduct  of  its  sine  and  a  unit 
vector,  at  right  angles  to  their  plane  and  such  that  the  rotation 
which  causes  the  divisor  to  coincide  in  direction  ivith  the  dividend 
shall  be  positive. 

26.   If,  for  ^  =  g,  we  write  '-  =  q[  it  is  evident  that  g'  differs 
(i  a- 

from  q  both  in  the  act  of  tension  and  ver- 
sion ;  tlie  tensor  of  q'  being  the  reciprocal 
of  the  tensor  of  g,  and  the  unit  vector  e, 
while  still  parallel  to  its  former  position, 
is  reversed  in  direction  (Art.  23)  since 
the  direction  of  rotation  is  reversed  (Fig. 
30) .     Hence 

^  =  ^(cos<^-esin0)      ....     (17). 

a       Ta 

'1  is  called  the  recijyrocal  of  -.     As  already  remarked,  the 
a  jB 

positive  direction  of  €  is  a  matter  of  choice.  It  is  only  neces- 
sary that  if  we  have  -f-  e  in  U-,  we  must  have  —  e  in  U  _,  or 
conversely. 


40 


QUATERNIONS. 


Fig.  31. 
J 


27.  Let  ?,  J,  k  (Fig.  31)  represent  unit  vectors  at  right  angles 
to  eiich  other.  The  eliect  of  any  unit  vector  acting  as  a  multi- 
plier upon  another  at  right  angles  to  it, 
has  l)een  defined  (Art.  24)  to  be  the 
turning  of  the  latter  in  a  positive  direc- 
tion in  a  plane  perijendicular  to  the  ope- 
rator or  multiplier  through  an  angle  of 
90?  Thus,  /  operating  on  ,/  produces  k. 
J.  This  operation  is  called  multijjlica- 

tion,  and  the  result  the  product,  and  is 
expressed  as  usual 


V=  ^ 


(18). 


The  quotient  of  tn^o  vectors  being  a  factor  which  converts 
the  divisor  into  the  dividend,  we  have  also 


k^  . 
J 


(10), 


either  the  product  or  quotient  of  two  unit  vectors  at  right  angles  to 
each  other  being  a  unit  vector  2wr pen dicular  to  their  i^lane. 

This  multiplication  is  evidently  not  that  of  algebra ;  it  is  a 
revolution,  which  for  rectangular  vectors  extends  thi'ough  90? 
Nor  is  k  in  Equation  (18)  a  numerical  product,  nor  i  in  Equa- 
tion (19)  a  numerical  quotient.  This  kind  of  multiplication  and 
division  is  called  geometric. 

In  accordance  with  the  above  definition  we  may  write  the  fol- 
lowing equations 


(20). 


ij  =  k 

k_  . 
J 

jk  =  i 

I 

ki=J 

i=k 

i 

ji  =  -k 

-k      . 

kj  =  —  i 

-''-k 

GEOMETBIC   MULTIPLICATION   AND   DIVISION. 


41 


^•(-0  =  -i         ^  =  ^  1-    .     .     .     .    (20). 
Jc(-j)  =  i 

Since  the  effect  of  i,  Jc,  J  as  operators  is  to  turn  a  line  from  one 
direction  into  another  wlaich  differs  from  it  by  90°  they  are 
called  quadrantal  versors. 


-J 

=  i 

-A: 

y 

-./ 

-k 

=  i 

—  j 

=  k 

—  i 

i 

=  k 

-J 

—  i 
-k 

=J 

k 

=j 

—  i 

28.   Since 

i  Xj  =  k     and     i  x  k  =  —  j  =  —1  X  j, 
we  have 

iXiXj=—lXJ, 
or 

i  X  I  =  —  1 . 

"We  may  denote  the  continued  use  of  i  as  an  operator  by  an 
exponent  which  indicates  the  number  of  times  it  is  so  used. 
This  is  consistent  with  the  meaning  of  an  exponent  in  algebraic 
notation.  In  both  cases  it  denotes  the  number  of  times  the 
operator  is  used,  in  one  instance  as  a  numerical  factor,  in  the 
other  as  a  versor.     Thus 

0  .0       ui      /'^        , 

jjm  =  j-i^^     _=-,     etc. 

JJ     r 
In  confonnity  to  this  notation  the  above  equation  becomes 

i'=-l (21), 


42  QUATERNIONS, 

aud  in  a  similar  manner, 


(22). 


Ti'r.  31. 
J 


Hence  the  square  of  a  unit  vector  is  — I. 

The  meaning  of  tlie  word  "  square  "  is  more  general  than  that 
which  it  possesses  in  Algebra,  as  was  that  of  tlie  word  "  product" 
in  Art.  27.  The  propriet}'  of  this  ex- 
tension of  meaning  lies  in  the  fact  that 
for  certain  special  cases,  the  processes 
above  defined  reduce  to  the  usual  alge- 
braic processes  to  which  these  tenns 
were  originally'  restricted.  The  condu- 
^^  sion  i-  =  —  l  is  seen  to  follow  directly 
from  the  definition,  since  if  i  operates 
twice  in  succession  on  either  ±  j  or  ±  k, 
it  turns  the  vector,  in  either  case  suc- 
cessively' through  two  right  angles,  so 
tliat  after  the  operation  it  points  in  the  opposite  direction.  A 
similar  reversal  would  have  resulted  if  the  minus  sign  had  been 
written  before  the  vector.  Thus  —  {±j)  =  T  j-  Hence  i  x  i, 
or  i-,  as  an  operator,  has  the  effect  of  the  minus  sign  in  revers- 
ing the  direction  of  a  line. 


29.  It  is  to  be  observed  that  so  long  as  the  cyclical  order  i,  j, 
A',  t,  J,  A;,  j,  ....  is  maintained,  the  product  of  any  two  of  these 
three  vectors  gives  the  third  ;  thus 


and  therefore 


as  also 


ij=k,    jk=i,     ki  =  j; 

{ij)k  =  kk  =  h~=-l, 
(jk)i=  a  =  r  =  — 1, 

{^OJ  =  JJ  =  f  =  - 1 ; 
i(jk)=  a  =  r-=  — 1, 

k{ij)  =AA-  =  /r=-l, 


GEOMETRIC   MULTIPLICATION   AND   DIVISION.  43 

hence 

iU^c)  =  {iJ)k, 

which  involves  the  Associative  laiv. 

"We  may  therefore  omit  the  parentheses  and  write 

yk^JJd  =  kij=-i (23), 

or,  the  continued  product  of  three  rectangidar  tinit  vectors  is  the 
same  so  long  as  the  cyclical  order  is  maintained. 
But 

kUi)=k{-l-)  =  -l'=l     ....     (24), 

or,  a  change  in  the  cyclical  order  reverses  the  sign  of  the  loroduct. 

30.  In  Equation  (24)  we  have  assumed  that 

A-(-A')  =  -K-. 

That  this  is  the  case  appears  from  the  fact  that  i  operating  on 
—  /  produces  —  A:,  or 

i{-j)  =  -k, 

and  that  the  same  result  would  be  obtained  b}'  operating  with  i 
on  J,  producing  yfc,  and  then  reversing  k.  That  is,  to  turn  the 
negative,  or  reverse,  of  a  vector  through  a  right  angle,  is  the 
same  as  turning  the  vector  through  a  right  angle  and  then  re- 
versing it.  Tlie  negative  sign  is,  therefore,  commutcdive  tvith  i, 
;,  k,  or 

^■(-i)  =  -0'  =  -^' (25). 

31.  It  follows  du-ectl}'  from  the  definition  of  multiplication, 
as  applied  to  rectangular  unit  vectors,  that  the  commutative  prop- 
erty of  algebraic  factors  does  not  hold  good.     For 

y  =  ^, 
but 

Ji  =  -k. 


44 


QUATERNIONS. 


Hence,  to  cJiange  the  order  of  the  factors  is  to  reverse  the  sign 
of  the  2)7'oduct.  The  operator  is  always  "oi-ittcn  first;  and,  since 
the  order  cannot  be  changed  without  affecting  the  result,  in 
reading  such  an  expression  as  ij  =  7c,  this  sequence  of  the  factors 
must  be  indicated  by  saying  "  i  into  j 
equals  7c"  and  not  "  i  multiplied  by  j 
equals  7i,"  the  latter  not  being  true. 

Hence  also  the  conception  of  a  quo- 
tient as  a  factor  requires  a  similar  dis- 
tinction, which  in  Algebra  is  unncces- 


Fi;:.  31. 


ji  =  k  is  not  true. 


sary.      In  the    latter,  from  -  =  a  we 

h 

have,  indifferentl}',  ah  =  c  and  ha  =  c. 

But  from  -  =  i,  while  iJ  =  k  is  true, 
J 
In  expressing  therefore  the  relations  be- 
tween i,  j  and  k  by  multiplication  instead  of  division,  care  must 
be  taken  to  conform  to  the  definition,  the  quotient  being  used 
as  the  multiplier  or  operator  on  the  divisor.  This  non-com- 
mutative propcrt}'  of  rectangular  unit  vectors,  which  results 
directl}'  from  the  primary  definition  of  the  operation  of  multipli- 
cation, will  be  seen  hereafter  to  extend  to  vectors  in  general 
and  to  quaternions,  whose  multiplication  is  not  commutative 
except  in  special  cases. 

The  quotient  then  being  a  factor  which  operates  on  the  divisor 
to  produce  the  dividend,  we  have 

^;./=A-,    that  is,    y^=k  ....     (26), 

the  cancelling  being  performed  by  an  upward  right-handed  stroke. 
But'^^  =  7i'  is  not  time,  for  this  would  involve  Jt=  ij. 


32.   It  follows  also  that  the  directions  of  rotation  of  a  fraction. 


as  -,  and  its  reciprocal  are  opposite. 
J 


Thus 


-=  • 


k 


(27), 


GEOMETRIC   MULTIPLICATION   AND   DIVISION.  45 

and  therefore  that  the  reciprocal  of  the  quotient  i  is  —  ^■,  or 

\--i (28); 

I 

that  is,  the  recij)rocal-of  a  unit  vector  is  the  vector  reversed.     Tliis 
maj'  be  written 

\^i-'^-i    ......     (29), 


the  exponent  denoting  that,  as  a  factor  or  versor,  i  is  used  once, 
wliile  the  minus  sign  before  the  exponent  indicates  a  reversal  in 
the  direction  of  rotation. 

33.    If  a  be  any  unit  A'ector,  we  obtain  from  the  preceding 
Ai'ticle 

a  —  =  a( —  a) 


But 


hence 


1    1 


-a:^a  — 


(30), 


or,  a  unit  vector  and  its  reciprocal  are  commutative  and   their 
product  plus  unity. 

If  a  is  not  a  unit  vector, 

a  =  TttUa, 

-=-^  =  -— Ua (31), 

the  tensor  of  the  reciproccd  of  a  vector  being  the  reciprocal  of  its 

tensor. 

k 
It  must  be  care  full}'  observed  that  a  fraction,  as  -,  cannot  be 

1        1  *' 

written  indifferenth^  k-  or  -/c,  for  this  would  involve  ki~^  =  i~^k, 

i       i 
which  is  not  true. 


46 


QUATERNIONS. 


B}'  definition  A' (—?')  =  —  /.  or  /./"'  =  k-  =  —/  =  -.     Hence, 

k        1  '                       i         '       i 

-  =  ]c-  or  ki  \  From  the  meaning  attached  to  the  ordinary' 
I         i 


notation  of  algebra, 


.A-      A- . 
i      i 


(a) 


would   appear  to  be   coiTect ;  for,  cancelling,  we  have  A;  =  A\ 

k  1 

"Whereas,  since  -  must  be  written  A;-,  we  should  have 

i  i 


or 


iki-^  [=  -  ikq  =  A-r'  /  [=  A-] 
jt[=-A-]=A;, 


which   is   not  true. 


Of  course   that   equation  (a)  is  false  is 
directl3'    evident    from    the    fact    that 

-7=  —j,  and  (a)  involves  i  (  — ./)  =  (  —j)  i 

or  ij  =ji.  The  above,  however,  shows 
that,  as  cancelling  must  be  performed 
by  an  upward  right-handed  stroke 
when  the  exi)ressiou  is  in  the  form  of 
a  quotient  or  fraction,  so  when  ex- 
pressed in  the  form  of  multiplication, 
the  cancelled  factors  must  be  adjacent. 
In  such  an  expression  as 


.; 


=  -./'■"  V"  =iy  =  -i 


it  might  be  supposed  permissible  to  write  also 


(&) 


(c) 


since  in  either  case  the  correct  result  is  obtained.  This  arises, 
however,  from  the  fact  that  both  the  fractions  in  the  first  mem- 
ber of  (6)  are  equal  to  A\  and  therefore  may  be  permuted  so  as 

to  read  kk  =  ^^yi^  =  ^^  =  —  1 .     The  process  of  (c)  is,  how- 


GEOMETRIC   MULTIPLICATIOISr   AND   DIVISION,  47 

ever,  illegitimate,  and  the  result  is  correct,  not  because  the 
process  is  so,  but  because  the  factors  are  in  this  case  commu- 
tative. 

34.    Since  the  act  of  tension  is  independent  of  that  of  version, 
and  their  order  is  immaterial, 

xi  .  yj  =  xy  .  ij  =  yx  .  ij  =  zk      ...     (32) , 

where  x  and  y  are  any  two  scalars  and  xy  =  z.  Hence  the  com- 
mutative principle  applies  to  tensors.  If  then  a,  f3,  y  are  in  the 
direction  of  i,  j  and  k  respectively,  and  a,  &,  c  are  theu'  tensors, 

a/3  =  TaT/?  .  ij  =  ab  .  k, 

ay  =  TaTy  .  ik  =  —  ac  ,  j,  etc., 

or,  the  product  of  any  two  rectangular  vectors  is  the  product  of 
their  tensors  and  a  unit  vector  at  right  angles  to  their  plane. 
So  also 

a  _  Ta  .  I  _  Ta        £  _  _  <^'  7. 

a      Ta  .  i       a  .      , 

-J,  etc., 


y      Ty .  k      c 

or,  the  quotient  of  two  rectangular  vectors  is  the  quotient  of  their 
tensors  times  a  unit  vector  at  right  angles  to  their  plane. 

35.   If,  as  above,  a  =  ai,  then 

aa  =  ai .  ai, 

a'  =  cv  i", 

a^  =  -a- (33). 

Hence,  tlie,  square  of  any  vector  is  minus  the  square  of  its  tensor. 
Since  Ta  =  a  is  the  ratio  of  the  lengths  of  a  and  TJa,  the  square 
of  any  vector  is  the  square  of  the  corresponding  line,  regarded  as 
a  length  or  distance  only,  with  its  sign  changed. 
If  ai  =  a  and  bi  =  (3, 

a/3  =  abi-  =  —  ab. 


48 


QUATERNIONS. 


36,   That  the  multiplication  of  rectangular  vectors  is  a  dis- 
tributive operation  maj'  be  seen 
directl}'  from   Fig.   32   by  ob- 
'^        servina:  that 


Fig.  3-2. 


iU  +  l-)=iJ  +  ik    (34), 

i  being  perpentlicular  to  and  in 
front  of  the  plane  of  the  paper. 


37.   Exercises  in  the  transformations  of  ?*,  j,  k 


1. 

J(-0-- 

=  k. 

2. 

Ji-k)  = 

3. 

k(-j)-- 

=  i. 

4. 

k(-i)  = 

5. 

-kU)-- 

=  i. 

G. 

(-k)i  = 

7. 

(-J)k- 

=  —  i. 

8. 

(-J)(-k)  = 

9. 

(-./)(- 

-0  = 

•10. 

(-0(-i)  = 

11. 

J    __ 

k. 

12. 

i 

13. 

-A- 

14. 

A- 

15. 

J 

-k 

JC. 

17. 

i 

18. 

19. 

20. 

k  _ 

21. 

k  _ 

22. 

1     A-_ 

23. 

ijk_ 

kji 

24. 

i     j     k  _ 

k  '  j  '  i 

25.  Is  it  correct  to  write,  in  general,  the  product  of  anj*  frac- 
tions, as  -  .  -,  in  the  form  —  ? 

./    J  _    JJ 

26.  State  whether  ^^—  .  -  =  — -  is  correct  or  not,  and  why. 

k       i        ki 

27.  i^fk'  =  -{ijJcy. 


GEOIMETRIC    MULTIPLICATION   AND   DIVISION.  49 

38.   Resuming  Equation  (15), 

a       Ta  /  ,    ,        •      , \ 

q—~  =  —  {cos<i  +  €Sin<i) ; 

the  quaternion  q  was  shown  (Art.  25)  to  be  the  product  of  a 
tensor  and  a  versor.     It  may  also  be  regarded  as  the  sum  of  two 

~Ta 
cos  dt 

T(3        ^_ 
that  of  the  cosine  of  tlie  angle  (^)  between  the  vectors,  while  the 

'^  -■--  '  is  a  vector  at  right  angles  to  their  plane. 


parts,  the  first  of  which 


is  a  scalar,  whose  sign  is 


second 


T/3 


sin  (f) 


whose  sign  depends  upon  the  direction  of  rotation  of  the  fraction 

(35), 


-.     This  ma}"  be  expressed  symbohcally  in  the  notation 


so  that  we  have  both 
and 


^      P         (3         {3 


q  =  TqVq 
q  =  Sq  +  Yq. 


The  second  member  of  this  last  equation  is  read  ' '  scalar  of  q 
plus  vector  of  q,"  ^q  and  Yq  being  respective!}'  s^nnbols  for  the 
scalar  and  vector  parts  of  the  quaternion.  As  ah-ead}'  explained 
in  the  case  of  the  sj'mbol  S,  V  is  a  symbol  of  operation,  denoting 
the  operation  of  taking  the  vector  terms  of  the  expression  before 
whicli  it  is  written. 

The  quotient  of  tivo  vectors  is,  therefore,  the  sum  of  a  scalar 
and  a  vector. 


The  scalar  of  the  quotient 


8(7  = COS  (h 

.  T/3 


is  the  ratio  of  the 


tensors  times  the  cosine  of  the  contained  angle.      The  tensor  of 


the  vector  part 


TVf/  =  —  sinci 


is  the  ratio  of  the  tensors  times 


the  sine  of  the  contained  angle.     The  versor  of  the  vector  part 
[UYg  =  e]  «s  a  unit  vector  perpendicular  to  their  plane.,  having  a 


50 


QUATERNIONS. 


direction  such  (hat  the  direction  of  rotation  of  the  divisor  is  posi- 
tive or  left-handed. 

Letting  a  and  b  be  the  tensors  of  a  and  /?,  and  collecting  the 
preceding  expressions  for  facility  of  reference,  we  have 


0 

Ug=cos<^  +  esin  (f) 

Sq=-  cos^ 
b 

\q=-  siu^  .  c 

TV9=-  sin<^ 
UTry  =  e 
SUg=cos  ^ 
VU7=sin  (^  .  € 
TVU(/=sin<^ 


(36). 


These  expressions  require  no  further  explanation  than  that 
derived  from  a  simple  inspection  of  Equation  (15)  in  connection 
with  the  meaning  akeady  assigned  to  T,  U,  S  and  V  as  symbols 
of  operation. 


39.   De  Moivre's  Pormula. 

The  following  considerations  will  explain  why  the  parenthesis 
(cos <^  + esin <}!))  as  a  versor  turns  P  left-handed  through  an 
angle  ^.  Tlie}'  also  contain  the  quaternion  inteqDretation  of 
imaginary  quantities. 

Let  -y  =  sin  <^  and  z  =  cos  <^. 

Differentiating, 


or 


dv  =  cos  <^  c?<^,     dz  =  —  sin  <^  d^, 

dv  =  zd<{>^ 
dz  =  —  vdtjj. 


(a) 


GEOMETRIC    JNIULTIPLICATIOX   AND   DIVISION.  51 

Multiplj'ing  (a)  b^-  V— 1,  and  adding  the  result  to  (6), 
dz  +  dv  .  V^=  {—v+z  -s/^l)  d<)>, 
dz  +  dv  .  V^  =  {v  V^^+  z)  V^  d(l>, 


whence 


d{z  +  v^/-l)  /— - 

——=-=  dcj>  .  V-  1,  (c) 

z  +  v V  — 1 

which  may  be  writteH     "' 

2  +  t;  V— 1=  e'>^~'; 
Ol- 
eosa +  sua  (/).  V—l=  e*^^^,  (cZ) 
whence 

cos  HI  (/)  +  sin  m^  .  V  —  1  =  e'"''^  ^~^.  (e) 

But  we  have  from  (d) 

(cos  (ji  +  sin  (^  .  V^  )  "» =  €'»'>  ^~i,  (/) 

and  therefore,  from  (e)  and  (/), 

(cos</)  +  sincj)  .  V  — 1)"*=  cos m(j>  +  sin  7?i<^  .  V— 1   (37), 

which  is  the  well-known  formula  of  De  ]Moi\Te. 

This  formula  may  be  made  the  basis  of  a  s^'stem  of  analytical 
trigonometry.  Thus,  for  example,  to  deduce  the  formulae  for 
the  sine  and  cosine  of  the  sum  of  two  angles,  we  have  from  (f?) 


cos  cji  +  sin  cj)V  —I 


Ji-Z^i 


cos  6  +  sin  (9  V  —  1  =  &■ 
Multipl^-ing  member  b}-  member, 

cos <^  cos 0  +  cos <^  sin ^  .  V  —  1  +  cos ^  sin  ^  .  V—  1  — 

sin</>sin^  =  e^^  +  ^^^^-  (S') 

But  from  De  Moivre's  formula 

cos  ((/.  +  ^)  +  sin  {cj>  +  e)\/^l=  6^'^  +  ^^^'-  ('0 


52 


QUATERNIONS, 


Equating  the  first  mombcrs  of  (f/)  and  (h),  since  in  any  equa- 
tion between  real  and  imaginary  (quantities  these  are  separately 
equal  in  the  two  members,  we  have 

cos  [0  -\-  cf))  =  cos  6  cos  <^  —  sin  ^  sin  ^. 
sin  {6  +  ^)  =  sin  0  cos  <^  +  cos  6  sin  ^. 


These  formulae,  while  they  ma^-  be  of  course  demonstrated 
independently  of  De  Moivre's  formula,  are  here  deduced  from 
imaginary-  expi'cssions.  It  would  therefore  appear  that  these 
expressions  admit  of  a  logical  interpretation. 

If  any  positive  quantity  7)i  be  multiplied  bj-  (V— 1)-  the  re- 
sult \s  —  m.  That  is,  in  accordance  with  the  geometrical  inter- 
pretation of  the  minus  sign,  we  ma^*  regard  the  above  factor 
( V— 1)"  as  having  turned  the  linear  representative  of  m  about 
the  origin  through  an  angle  of  180?  If,  instead  of  multiplying 
m  by  ( V  —  1 ) ",  we  multipl}-  it  by  V— 1,  we  ma}'  infer  from 
analogy  that  the  line  m  has  been  turned  through  an  angle  of  90° 

about  the  origin.  If,  too,  we  ob- 
serve that  each  of  the  four  expres- 
sions 


Fi!?.  33. 


}' 

-<•* 

"t  v^  \ 

x'i  -w 

0      'II, 

1-^' 

1 
\ 

\ 

^ 

m,     m 


V-1,     -1 


—  ?u  V— 1 


is  obtained  from  the  preceding  by 
multi])hing  b}' the  factor  V  —  1 ,  they 
may   be    regarded   as   denoting    in 
order  a  distance  m  on  the  co-ordi- 
nate   axes    OX,    OY,    OX,'    OY' 
(Fig.  33),  V— 1  being,  as  a  factor, 
a  versor  turning  a  line  left-handed  through  a  quadrant.     These 
expressions  therefore  locate  a  point  on  the  axes,  both  as  to  dis- 
tance and  direction  from  the  origin. 

Since  ever}-  imaginar}-  expression  can  be  reduced  to  the  form 
±  a  ±b  V  — 1,  we  ma}',  in  accordance  with  the  above  interpre- 
tation of  V  — 1,  regard  such  an  expression  as  defining  the  posi- 
tion of  a  point,  out  of  the  axes.     Thus  oa  =  a  (Fig.  34)  and 


GEOMETRIC   MULTIPLICATION   AND   DIVISION. 


53 


AP  =  &,  laid  off  at  a  at  right  angles  to  oa  since  h  is  multiplied 
b}'  V  —  1  ;  so  that  in  passing  over  oa  and  ap  in  succession  we 
reach  the  point  p.  It  is  also  evident  that  such  an  expression 
Implicitl}'  fixes  the  position  of  p  b}' 
polar  co-ordinates,  since  Va-  -f  b"^  =  op 

and    tan  poa  =  - .       In    like    manner 
a 

—  6  +  a  V  —  1  would  locate  a  point  p^ 
oa'  having  a  length  =  a,  but  laid  off 
perpendicular  to  oa,  since  V  — 1  is  a 
factor,    and   a'p'=  —  h.      As   before, 

we  have  implicitl}^  op'=  Va^-f-&"  and 

,         ,  a 

tan  p  OA  =  — . 


T^'l 

Fig 

'.34. 

/ 

o 

A 

Furthermore,  if  we  operate  on  the 
first  expression,  a+6V  — 1,  which  fixes  the  point  p,  with 
V— 1,  we  obtain  the  second,  —  6  +  aV  — 1,  or  V— 1  as  a 
factor  turns  op  thi'ough  90°  so  as  to  make  it  coincide  with 
opI  As  an  operator,  therefore,  we  may  regard  V— 1,  like  i,  J, 
Zj,  as  a  quadrantal  versor,  tmniing  a  line  through  a  quadrant 
in  a  positive  direction.  Algebraically  it  denotes  an  impossible 
operation.  (In  Algebra  quantities  are  laid  off  on  the  same 
line  in  two  opposite  directions,  +  and  — .  It  was  because  quan- 
tities are  so  estimated  onl}-  in  Algebra  that  Su-  W.  Hamilton 
called  it  the  Science  of  Pure  Time,  since  time  can  be  estimated 
onl}-  into  the  future  or  the  past.)  But  it  is  unreal  or  imaginary 
only  in  an  algebraic  sense.  If  the  restrictions  imposed  by  Al- 
gebra are  removed,  b}'  enlarging  our  idea  of  quantit}'  and  at  the 
same  time  modifj'ing  the  operations  to  which  it  is  subjected,  this 
imaginary  character  disappears.  In  applying  the  old  nomen- 
clature to  these  new  modifications,  it  will  be  seen  that  the  prin- 
ciple of  permanence  is  observed,  i.e.,  the  new  meaning  of  terms 
is  an  extension  of  the  old  ;  and  when  the  new  complex  quantities 
reduce  to  those  of  Algebra,  the  new  operations  become  identical 
with  the  old. 

If  now  we  operate  upon 

a-i-6  V  — 1, 


54 


QUATERNIONS. 


which,    if    we   regard   a  =  oa  (Fig.    35)    and   ^V  — 1=  ap   as 
vectors,  is  equivalent  to  or,  with  the 
Fig.  35.  expression 

cos<^  +  sin</)  .  V— 1 
of  De  Moivre's  formula,  we  obtain 


a  cos  <^  —  Z>  sin  <^  +  V  —  1  (a  sin  4>  + 
h  cosff)). 

Draw  OX'  so  that  X'OX=cf>;  also 
pa"  and  ai.  perpendicular  and  as  par- 
allel to  ox:     Then 

a  cos  <f>  —  b  sin  <^  =  ol  —  a"l  =  oa" 
a  sin  <^  +  6  cos  (f>  =  la  4-  sp  =  a"p. 


-'X' 


Make  oa'=  oa"  and  lav  off  a'p'=  a"p  perpendicular  to  OX, 
since  it  has  V  —  1  as  a  factor  ;  then 

(a  cos  <^  —  &  sin  <^)  +  V  —  1  (a  sin  <j>  +  b  cos  (f>)  =  o\'+  a'p'=  op,' 

and  p'op  =  <l>. 

But  the  formulae  for  passing  from  a  set  of  rectangular  axes 
OX,  0  Y,  to  another  rectangular  set  OX',  0  Y',  are 

X  =  x'cos  </)  +  y'sin  (f>, 
y  =  y'coscf>  —  x'sincf>, 

in  which  A''OX'=  </>,  .i;  =  oa,  y  =  xv,  x'=  o\','  y'=vx','  or 


OA  =  OK  +  KA, 
AP  =  NP  —  a"k, 


a"k  being  perpendicular  and  a"x  parallel  to  OX. 

Hence  the  effect  of  the  operator  has  been  to  turn  op  left- 
handed  through  an  angle  </>,  which  is  equivalent  to  turning  the 
axes  rio-ht-handed  thi-ousrh  the  same  angle. 


GEOMETRIC   MULTIPLICATION   AND   DIVISION. 


55 


+  1,-1  and  V  — 1  are  particular  cases  of  the  general  versor 

cos  ^  +  sin  <^  .  V  —  1 , 

nameh',  when  cf>  is  0°  180°  and  90°  respectively,  +  1  preserv- 
ing, —  1  reversing  and  V— 1  semi-inverting  the  line  operated 
upon. 

"We  may  now  see  the  meaning  of  De  Moivre's  formula 

(coscf)  -f  sinc^  .  V— 1)™  =  cosmcf)  +  sin  mcfi  .  V  — 1. 

As  operators,  the  first  member  turns  a  line  through  an  angle  ^ 
successively  m  times,  while  the  second  member  turns  it  through  m 
times  this  angle  once,  producing  the  same  result.  The  expressions 
cos  <f)  +  sin  <^  .  V  — 1  and  cos  ^  +  sin  <^  .  e  are  identical,  except 
that  in  the  latter  the  plane  of  rotation  is  not  indeterminate, 
being  perpendicular  to  e,  V— 1  being  any  unit  vector  loitli  in- 
determinate direction  in  sjjace. 

Equation  (37)  may  be  put  under  the  form 

cos  m  (2  irn  -f  ^)  +  sin  m  (2  tth  +  ^)  .  V  —  1  =  [cos  (2  ttw  +  <^)  + 
sin  (27rn-|-  <^)  .  V  — 1]™. 

In  the  second  member  if  <^  =  0  and  m  =  ^,  we  have  Vl  for  all 
integral  values  of  ?i,  while  the  first  member  for  w  =  0,  n  =  1 , 
n  =  '2  becomes  1,  -i  +  ^^V^^l,  -^-^V^,  the  three 
roots  of  unity. 

In  the  same  wa}'  for  m  =  ^, 

1, 

i  +  -fV-i_, 
-1,    _  


^r= 


the  six  roots  of  unity.     The  real  roots  lie  on  the  axis,  along 
which  du-ection  is  assumed  plus  and  minus,  while  the  imaginary 


56 


QUATERNIONS. 


roots  arc  vectors  in  a  direction  not  that  of  the  axis,  and  are  the 
sum  of  two  vectors,  one  of  which  is  in  the  direction  of  the  axis 
and  the  other  perpendicular  to  it. 

40.  Let  a  and  (S  be  unit  vectors  along  oa  and  or  (Fig.  3G). 
Resolve  oa  =  a  into  the  two  vectors 
CD,  DA.     Then 


Fig.  36. 


OA  =  a  =  CD  -|-  DA. 


But 


OD  =  cos  (f>  .  (3, 
DA  =  c  (sin  (fi  .  13)  =  sin  <j>  .  €/5, 

6  being  a  unit  vector  perpendicular  to 
the  plane  aob,  as  in  the  figin-e.    Hence 


a  =  cos  (f>  .  (3  -{-  sin  (f>  •  «/?• 


(^0 


Now  when  a  and  (3  are  luiit  vectors,  we  have  bj'  definition 

^  .  f3  =  (cos  4>-\-  (.  sin  (f>)(3  =  a  ;  or,  comparing  with  (a) , 

(cos ^  +  csiu 4>)^  =  cos </)./?  +  sin ^  .  e^. 

Tlie  clistributive  law,  therefore,  applies  to  the  multiplication 
of  a  vector  by  the  scalar  and  vector  ixtrts  of  a  qi(aternion;  for 
if  a  and  /?  are  not  unit  vectors,  the  tensors,  as  merely  numerical 
factors,  can  be  introduced  without  affecting  the  versor  conclu- 
sion. Resolve  ft  into  the  vectors  oc,  cb,  cb  being  perpendicular 
to  OA.     Then 

OB  =  /3  =  oc  +  CB. 


But 

Hence 


oc  =  cos  <^  .  a,      CB  =  —  c  (sin  (/>  .  a) . 
cos  0  .  a  —  sin  ^  .  ca  =  yS, 
or,  by  the  distributive  principle, 

(cos  (^  —  sin  <^  .  e)  a  =  (3. 


GEOirETEIC   ISrULTIPLICATIOX   AND   DIYISIOX.  57 

Using  the  two  memlDers  of  this  equation  as  multipliers  on  the 
corresponding  members  of  («) 

(cos  ^  —  sin  (^  .  e)  aa  =  /?  (cos  (^  .  ^  +  siu  <^  .  e/?) , 

or,  since  a-=  —  1, 

—  cos  <;?)  +  €  sin  ^  = /5a (38). 

If  a  and  /?  are  not  unit  vectors, 

/Sa  =  T/3Ta(-cos<^4-esin<^)      .     .     .    (39). 

Operating  with  each  member  of  (a)  on  yS, 

ap  =  (cos  <^  .  /3  +  sin  (^  .  e/3)y3 
=  cos^  .  /3"+  sin<^  .  e/3' 
=  — cos^  — esin<^ (40), 

or,  if  a  and  /3  are  not  unit  vectors, 

a^=TaT;S(— COS(/)  — esinc^)       .      .      .      (41). 

Tlie  product  of  any  tivo  vectors  is,  therefore,  a  quaternion, 
■which,  as  before,  ma3'  be  regarded  either  as  the  sum  of  a  scalar 
and  a  vector  or  the  product  of  a  tensor  and  a  versor.  In  gen- 
eral notation 

aft  =  Saf3 +Yal3  =  Hq +Tq    ....      (42), 
al3  =  Tq.Vq (43). 

TJie  scalar  of  the  product  [Sa/?  =  —  TaT^S  cos  ^]  is  the  product 
of  the  tensors  and  the  cosine  of  the  supplement  of  the  contained 
angle. 

Tlie  vector  of  the  prodixt  [Tay8  =  —  TaT/3  sin  0  .  c]  has  for  its 
tensor  [TVa/5=  TaT/3  sin  </>]  the  product  of  the  tensors  and  the 
sine  of  the  contained  angle,  and  for  a  versor  [tJVa/5  =  — c]  a 
unit  vector  at  right  angles  to  their  plane  such  that  rotation  about 
it  as  an  axis  is  positive  or  left-handed. 


68 


QUATERNIONS. 


Representing  the  tensors  of  a  and  fi  hy  a  and  6,  we  have,  as 
in  Art.  38,  from  Equation  (41), 


Fig.  36. 


Tqz=ab 

Vq  —  —  cos  (^  —  €  sin  </) 
Sg  =  —  ah  cos  <^ 
V^  =  —  ah  sin  ^  .  c 

TV^  =  ah  sin  «^ 

1  Try  =  -  € 

Sl'5  =  —  cos  </) 

VlVy  =  —  sin  ^  .  c 
TVUr/  =  sin  <;() 
(TV  :  S)(/  =  -tan<^ 


1-  (-t-i)- 


41.   Resuming  the  expressions  for  the  products  and  quotients 
of  a  and  ^, 

/Sa  =  T/3Ta(— cos  <^  +  c  sin  <;()),  (a) 

a^  =  TaT/?(-cos0-£sin0),  (6) 

(0 


C  =  ^('cos.^-esin(^), 
a        la 


we  observe 


a       Ta  /  J    ,        •     ,\ 

-  =  —  (cosd)  +  €sm<i), 


(d) 


1st.  That  if  a  and  ft  be  interchanged  the  sign  of  the  vector 
part  is  changed.  It  is  equivalent  to  a  reversal  of  the  angle  (f>, 
and  consequently  a  change  in  the  direction  of  rotation.     Hence 


UY^a=€  =  -rVa/5-\ 

uv^  =.  =  -rv?  [ 

ft  a    ) 


.     .     (45). 


Vector  multiplication  is  not  therefore  in  general  commutative. 
2d.    If  the  vectors  are  unit  vectors, 


a  ft 


.    (46), 


GEOMETRIC   MULTIPLICATIOX    AND   DIVISION.  59 

the  product  being  expressed  also  by  a  quotient.  This  is  of 
course  always  possible,  as  appears  from  (a),  (6),  (c)  and  (t?), 
and  the  transformation  may  be  efiected  thus  : 

^  =  -^  =  ^(eosc^-.sin<^),        [Eq.  (.31)] 

a  la         la 

—  (3a  =  T,8Ta  (cos  c/)  -  e  siu  <^)  ; 
or 

/3a  =  T/3Ta  (  -  cos  (^  +  e  siu  4,)  . 

3d.  If  ^  =  0,  then  in  either  (a)  and  (6)  or  (c)  and  (d) 
the  vector  part  of  q  becomes  zero,  and  the  quaternion  de- 
grades to  a  scalar.     When  <^  =  0  the  vectors  are  parallel,  and 

aB  = —  TaTB  =  —  ab,   as  in  Ai't.   35;    also  -  =  —  =  -,   as  in 

/3      T/3      b\ 

Art.  8.     If  at  the  same  time  a  and  (3  are  unit  vectors  -  =  - =1 

/3      a 
[or  =  aa ~^  =  —  a-  =  1]  and  a^  =  a-  =  —  1 ,  as  in  Arts.  33  and  28. 
If  then  q  he  any  quaternion  and  \q  =  0,  the  vectors  oftchich  q 
is  the  quotient  or  jiroduct  are  parallel. 

4th.  If  c/)  =  90°  then  in  either  (a)  and  (&)  or  (c)  and  {d) 
the  scalar  part  of  q  becomes  zero,  and  the  quaternion  degi-ades 
to  a  vector ;  and  either  the  product  or  quotient  of  two  rectangu- 
lar vectors  is  therefore  a  vector  at  right  angles  to  their  plane, 

a/3  reducing  to  —  abe  and  -  to  -c,  as  in  Art.  3-4.     If  at  the 
'  ^  (3        b 

same  time  a  and  /3  are  unit  vectors,  a/3  =  —  e  and  -  =  e,  as  in 
Art.  27.  ^ 

If  then  q  he  any  quaternion  and  Sg  =  0,  the  vectors  ofzchich  q 
is  the  quotient  or  product  are  perpendicular  to  each  other. 

5th.  If  an  equation  involves  scalars  and  vectors,  the  vector 
terms  having  been  so  reduced  as  to  contain  no  scalar  parts,  then 
since  the  scalar  terms  are  purely  numerical  and  independent  of 
the  others,  the  sums  of  the  scalars  and  vectors  in  each  member 
are  separately  equal.     Thus  if 

X  +  aa  -j- b/3  =  d  +  y -{-  a'a  +  (&'-  b")/3      ) 
then  C      (47), 

x  =  d  +  y     and     aa  +  h/3  =  a'a  +  (&'-  b")(3  ) 


60  QUATERNIONS. 

which  might  also  be  written  (i\j't.  38) 

S  (x  +  aa  +  6/3)  =  S  [rf  +  2/  +  a'a  +  (b'-  b")^] , 
y(.v.  +  aa  +  b/3)=\[cl  +  y  +  a'a  +  {b'-  &")/3]. 

Gth.   ^  being  the  quotient  which  operates  on  a  to  produce  ^, 
a 

we  have  by  definition  p 

j/=(3 (48). 

7th.  TVa^,  or  ab  sin<^,  is  the  area  of  a  paraUelogram  whose 
sides  are  equal  in  length  to  a  and  b  and  parallel  to  a  and  (3. 
Sa/3,  or  —ab  cos cf),  is  numerically  the  area  of  a  i)arallelogi'am 
whose  sides  are  a  and  6,  and  angle  ab  is  the  complement  of  <^. 

8th.  Since  the  scalar  s^-mbol  S  indicates  the  operation  of 
taking  the  scalar  terms, 

Sa  =  0 (49), 

and,  for  a  similar  reason, 

Ya  =  a (50) . 

Again,  since  a-  is  a  scalar, 

V(a2)=0 (51), 

8{a^)  =  -a- (52). 

T(a^)  ma}-  be  wTitten  T .  a^,  as  also  S(a^)  =  S  .  a^,  but  these  forms 
must  be  distinguished  from  (Va)-  and  (Sa)^,  which  latter  are 
also  sometimes  written  Va  and  S'a. 

9th.    Comparing  (a)  and  (6), 

Sa/3=S/3a (53), 

and 

Ta/3  =  -T/3a (54). 

Adding  and  subtracting  (a)  and  (b) ,  we  have  also 

a/3  +  /3a=2Saft (55), 

a/3-(3a=:2\a/3 (56). 


GEOMETRIC   MULTIPLICATION   AND   DIVISION.  61 

10th.  a/3  .  /3a  =  (Sa/3  +  Va/3)  (Sa/3  -Ya(3)    [Eqs.  (53)  and  (54)] 

=  {Sa/Sy-  Sa/3Va/3  +  Sa/3\a/3  -  (Va/?)^. 
HsucG 

a/3.  /3a  =  (Sa/3)2-(Va,8)-     ....      (57), 

or,  from  Equation  (44), 

a^  ./3a  =  (Ta/3)2 (58). 

42.  Powers  of  Vectors. 

The  symbol  i",  m  being  a  positive  whole  number,  has  been 
seen  (Art.  28)  to  represent  a  quadrantal  versor  used  m  times 
as  an  operator  ;  the  exponent  denoting  the  number  of  times  i  is 
used  as  a  quadrantal  versor.     B}'  an  extension  of  this  meaning 

of  the  exponent,  «™  would  naturally  represent  a  versor  which, 

1 
as  a  factor,  produces  the  — th  part  of  a  quacbantal  rotation. 

Thus  /^  produces  a  rotation  through  one-third,  and  i^  through 
three-fifths  of  a  quadrant,  respectively.  With  the  additional 
meaning  attached  to  the  negative  exponent  (Art.  32),  as  indi- 
cating a  reversal  in  the  direction  of  rotation,  we  may  in  general 
define  i',  where  i  is  any  vector-unit  and  t  any  scalar  exponent, 
as  the  representative  of  a  versor  lohich  ivoukl  cause  any  right 
line  in  a  plane  j)erj)endlcular  to  i  to  revolve  in  that  ^jlane  through 
an  angle  t  X  90°  the  direction  of  rotation  depending  upon  the 
sign  of  t.  Hence  ever}^  such  power  of  a  unit  vector  is  a  versor, 
and,  conversel}',  every  versor  may  he  represented  as  such  a 
poicer.  ^  2./> 

Since  the  angle  (^)  of  the  versor  is  ^  x  -,  we  have  t  =  — , 
and  any  versor 

cos  cji-\-  e  sin  (f) 
may  be  expressed  2« 

cos  (j!)  +  e  sin  ^  =  e"^ (59), 

and 

cosc/)  —  esin^  =  e"-^      ....     (00), 

the  vector  base  being  the  unit  vector  about  which  rotation  takes 
place,  and  the  exponent  the  fractional  part  of  a  quadi'ant  tlu'ough 
which  rotation  occurs. 


62  QUATERNIONS. 

The  operation  of  which  *-  is  the  agent  is  one-half  that  of 
which  i  is  the  agent,  and  therefore  two  operations  with  the 
former  is  eqnivalent  to  one  with  the  latter ;  or,  as  in  Algebra, 

iW^=i=i'^^'^ (Gl), 

or,  emplo3'ing  the  other  versor  form,  if  a,  yS,  y  are  complanar  nuit 
vectors  so  that 

a  .  2^ 

-  =  cos  ^  +  e  sm  <^  =  e  - , 


then  since 


we  have 


/?  2d 

-  =  cos  6  +  e sin  6  =  e~i 
7 

a      /3       a 


(cos  (^  +  e  sin  (f>)  (cos  9-{-  e  sin  0)  =  cos  4>  cos  9  +  t"^  sin  (^  sin  6  + 

€(sin<^cos^  +  cos  ^  sin  6) 

=  cos  (<^  +  $)+  £sin  ((/)  +  ^) . 

The  second  member  is  the  U— ,  its  angle  being  (<f>+0),  and 
ma}'  be  therefore  expressed  as  the  power  of  a  unit  vector,  and 

2(0+0) 

wntten  e — tt — 
the  factors,  or 


written  e — tt — ;  this  exponent  is  the  sum  of  the  exponents  of 


20    20  2(0  +  6) 

€^e~  =  €~^^~ (62). 


This  is  evidently  an  abridged  form  of  notation  to  which  the 
algebraic  Imu  of  indices  is  applicable. 

Since  £^=  —  1  and  therefore  €''=1  ;  if  £'=—1,  t  must  be  an 
odd  multiple  of  2,  and  if  £'=+1,  t  must  be  an  even  multiple 
of  2.  ^ 

In  either  case  the  coefficient  of  tt  in  ^  =  -tt  is  a  whole  num- 
ber, and  cos  <^  ±  €  sin  </)  degrades,  as  above,  to  the  scalar  ±1, 
since  sin  mtt  =  0  when  m  is  an  integer. 

If  e*  =  ±  €,  t  must  be  an  odd  number ;    in  which  case  also 


GEOMETRIC   MULTIPLICATION   AND   DIVISION.  63 

m=^,  f,  -^,  etc.,  cos  iJiTT  =  0  and  the  A'ersor  degrades  to  the 
vector  ±e. 

If  the  vector  is  not  a  unit  vector,  as  xi  =  p,  to  interpret  the 
exponent,  say  p5,  so  as  to  satisfy  the  formula 

P^p^  =  p (63), 

which  is  analogous  to  Equation  (61),  we  must  combine  with  the 
conception  of  rotation  through  half  a  quadrant  an  act  of  tension 
represented  by  the  square  root  of  the  tensor  of  p.  Thus,  if 
a;  =  16,  and  we  write 

p^  =  (16  0'=16^i% 
then 

pV  =  (16^r)  (16^ i^)  =  16 i  =  p, 

or,  if  a;=  Vs, 

p^  =  ^8  .  ?:^  =  V2  .  P, 

pW  =  ( V2  .  i')  ( V2  .  P)  ( V2  .  P)  =  V8  .  i  =  p. 

And,  in  general, 

p' =  (xiy  =  xf .  i' (64), 

or  the  tensor  of  the  power  is  the  power  of  the  tensor,  and  the 
versor  of  the  power  is  the  power  of  the  versor.     Symbolically 

T.p'  =  (Tpy (65), 

V,p'  =  {\]py (66). 

Any  such  power  (p') ,  as  the  representative  of  the  agent  of 
both  an  act  of  tension  and  version,  is  therefore  a  quaternion, 
whose  tensor  and  versor  can  be  assigned  by  the  above  rules,  and, 
conversely,  every  quaternion  can  be  expressed  as  the  power  of  a 
vector^  which  quaternion  may  degrade  to  either  a  scalar  or  a 
vector  as  seen  in  the  preceding  versor  conclusions.  Hence  it 
follows  that  the  index-law  of  Algebra  is  applicable  to  the  powers 
of  a  quaternion. 


G4  QUATERNIONS. 

43.   Relation  between  the  Vector  and  Cartesian  deter- 
mination of  a  point. 

If  «,  j,  k  are  three  unit  vectors  perpendicular  to  each  other  at 

a  common  point,  then  the  vector  from  this  point  to  any  point  p 

may  be  written 

p  =  xi  +  yj+zJc (07), 

in  which  x,  ?/,  z  are  the  Cartesian  co-ordinates  of  p.  If  the  vec- 
tors are  not  mutuall}'  perpendicular  and  are  represented  b}-  a,  /?, 

■y,  then 

p  =  xa  +  y(3-\-zy (G8), 

in  which  x,  ?/,  z  are  the  Cailesian  co-ordinates  of  p  referred  to 
the  oT)lique  axes.  So  long  as  the  vectors  a,  ^,  y  are  not  com- 
planar,  p  refers  to  any  point  in  space. 

Since  an}-  quaternion  q  ma}-  be  expressed  as  the  sum  of  a  sca- 
lar and  a  vector,  if  w  be  any  scalar,  then 

q  =  iv-\-  xa  +y(i  +  Zy (HO) . 

As  composed  of  four  terms,  we  observe  ah  additional  reason 
for  calling  this  complex  expression  a  quaternion. 
Any  vector  equation 

p  =  o-  =  Oa  +  i»^  +  Cy, 

involves  three  numerical  equations,  as 

x  =  a,    y=h,    z  =  €<, 
unless  the  vectors  are  complanar ;  in  which  case  we  may  write 

y  =  ?la  -f  m^, 

and 

p  =  {x  +  zn)  a  -f  (?/  -f  zm)f3, 
a-=  (a  +  en)  a  -f  (Z>  +  cm)/3^ 

which,  for  p  =  a,  involves  l)ut  two  equations 

x  +  zn  =  a  +  cn,     y-\-zm  =  h  -\-cm. 


GEOMETKIC   MULTIPLICATION   AND   DIVISION. 


65 


Resuming  the  quadrinomial  form  of  g,  when  the  component 
vectors  are  at  right  angles,  we  have 


q  =  zv  +  xi  +  yj+  zk 
Sg  =  10 
Yq  =  xi  +  yJ-\-  zk 


(70). 


Since  (TYq)-  =  -  (YqY  =  x^ -{-  if  +  z\  we  have 

TVg  =  Var  +  y-  +  z^  ] 


VYq. 


Yq        xi  +  yj-Jr  zk 


(71). 


TVg       Var  +  ?/2  +  2-'  J 
Also,  since  (Art.  41,  10th.) 

(Tqy  =  (Sg)^  -  {YqY  =  vr  +  .^^  +  f  +  z\ 


Ug 
SUg 

TVUg 


Tg  =  Vif-  +  .x-2  +  2/2  +  ^2 

g  w  +  xi  +  2/J+zfc 


Tg        Vtc^  +  of'  +  ?/-  +  z^ 
Sg    _  ?o 

Tg         Ay.j(;2  _j_  3^2  _^  2/^  +  2^ 
TVg^     I     a.-^  4-^24- ^2 

Tg  \^(;2_,_^.2_^^2_^^2    J 


(72). 


44.  The  plane  of  a  quaternion  has  been  alreach'  defined  as  the 
plane  of  the  vectors  or  a  plane  parallel  to  them.  The  axis  of 
a  quaternion  is  the  vector  perpendicular  to  its  plane,  and  its 
angle  is  that  included  between  two  co-initial  vectors  parallel  to 
those  of  the  quaternion.  If  this  angle  is  90°  the  quaternion  is 
called  a  Right  Quaternion.  Any  two  quaternions  having  a 
common  plane,  or  parallel  planes,  are  said  to  be  Complanar. 
If  their  planes  intersect,  the}'  are  Diplanar.  If  the  planes  of 
several  quaternions  intersect  in,  or  are  parallel  to,  a  common 
line,  they  are  said  to  be  CoUinear.  It  follows  that  the  axes  of 
coUinear  quaternions  are  complanar,  being  perpendicular  to  the 
common  line.     Complanar  quaternions  are  alwaA's  coUinear,  and 


66  QUATERNIONS. 

complanar  axes  coiTospoud  to  oollincar  quaternions,  but  the  lat- 
ter ma}'  of  course  be  tliplanar. 

O '  A  O "c 

Let  — -  and be  any  two  quaternions.     If  coinijlanar,  they 

o'b  o"d  -  1  I  ^        J 

ma}'  be  made  to  have  a  common  plane ;  and,  if  diplanar,  their 
planes  will  intersect.  In  the  former  case  let  oe  be  au}'  line  of 
their  common  plane,  or,  in  the  latter,  the  line  of  intersection 
of  their  planes.  Now,  without  changing  the  ratios  of  their  vec- 
tor lengths,  the  planes,  or  the  angles  of  the  given  quaternions, 
two  lines,  of  and  og,  may  always  be  found,  one  in  each  plane, 
or  iu  their  common  plane,  such  that  with  oe  we  shall  have 


O  A        OF  ,      O  C  _  OG 

O'b  ~~  OE  o"d  ~~  OE 


and,  therefore,  any  two  quaternions,  considered  as  geometric 
fractions,  can  be  reduced  to  a  common  denominator ;  or,  in  the 
above  case 

o'a      o"c  _  of      og  _  of  +  og 

O'U        o"d        oe        OE  ~         OE 

Moreover,  a  line  on,  in  the  plane  ao'b,  may  always  be  found 
such  that 

o'a  _  OE 

o'li      on 
and  therefore 

o"c     o'a_og    oe      og 
o"d     o'b  ~~  OE     on  ~  oh' 
and 

o'a  ^  o"c  _  of  ^  OG     of     oe_of 
o'b  '  o"d     oe  *  OE     oe    og     og 


45.  Reciprocal  of  a  Quaternion. 

The  reciprocal  of  a  scahir  is  another  scalar  with  the  same 
sign,  so  that,  as  in  -tUgebra,  if  x  be  au^'  scalar,  its  reciprocal  is 


GEOMETRIC   MULTIPLICATION   AND   DIVISION. 


67 


The  reciprocal  of  a  vector  lias  been  defined  (Art.  33) ,  so  that, 

if  a  be  any  vector,  -  =  a"^  = Ua. 

a  Ta 

The  reciprocal  of  a  quaternion  has  also  been  defined  (Art. 

26)  ;  thus  ^ 


being  any  quaternion. 


tt       q 


Fig.  37. 


is  its  reciprocal.     The  onl}-  difference  between  the   quotients 

-  and  ^  (Fig.  37)  is  that,  as  opera- 
^  a 

tors,  one  causes  ^  to  coincide  with  a, 
while  the  other  causes  a  to  coincide 
with  /3.  A  quaternion  and  its  recipro- 
cal have,  therefore,  a  common  plane 
and  equal  angles  as  to  magnitude, 
but  opposite  in  direction ;  that  is,  ^>^- 
their  axes  are  opposite.     Or 


Since 


Z  t  =  Z.q 
(1 


and     axis 


—  axis  q. 


and 


the  20Toduct  of  tivo  reciprocal  quaternions  is  equal  to  positive 
unity,  and  eacJi  is  equal  to  the  quotient  of  unity  by  the  other ; 

we  have,  therefore,  as  in  Algebra,  _g=l  and  ^'  =  7,  and  no 

g  1       '1 

new  symbol  is  necessary  for  the   reciprocal.      -  is,  however, 

sometimes  written  Rg,  R  being  a  general  symbol  of  operation, 
namely,  that  of  taking  the  reciprocal.  It  follows  from  the  above 
that 


T 


1 


1 

Tg 


^3), 


68 


QUATERNIONS. 


or,  the  tensors  of  reciprocal  qtiaternions  are  reciprocals  of  each 
other  ;  while  the  versors  differ  only  in  the  reversal  of  the  angle. 
If  then 


Ta 


we  shall  have 


<7  =  -  =  —  (cos  <i  +  e  sin  di) 


R.7  =  1  =  <7-i  =  ^  =  ^  (cos  <^  -  €  sin  <^) 


Ta 


[      '     (74). 


46.  Conjugate  of  a  Quaternion. 

If  ;S'  (Fig.  37)  be  taken  complanar  with  ^  and  a,  and  making 

with  a  the  same  angle  that  (3  does, 

Fig.  37.  T/3'  being  also  eqnal  to  T/?,  then,  if 

-  =  7,  — ,  is  called  the  conjugate  of 
/3      ^    ft' 

rj,  and  is  written  K7.     The  sjTnbol  K 
indicates  the  operation  of  taking  the 
conjugate.    A  quaternion  and  its  con- 
,>^''Sa  jiig'ite    have,    therefore,    a    common 

plane  and  tensor,  as  also,  in  the  ordi- 
nary' sense,  equal  angles  ;  but  their  axes  are  opposite  ;  or 


1 


and 
If  then 

we  shall  have 


Z  K7  =  Z  ry  =  Z 

TK7  =  1q  =  \ 
T? 

axis  K7  =  —  axis  q  =  axis 


fn 


a       Ta  /  ,     ,         ■      ,  \ 

q  =  -  =  —  (cos  d>  +  c  sm  <b) 

Ta 

Kg  =  — (cos(^-€sin<^) 


(75), 


(76), 


or,  the  tensors  of  conjugate  quaternions  are  equals  and  the  versors 
differ  only  in  the  reversal  of  the  angle. 

Regarding  a  scalar  and  a  vector  as  the  limits  of  a  quaternion 


GEOMETRIC   ]MTJLTIPLICATIO]Sr   AND   DIVISION.  69 

(Art.  41,  3d  and  4tli),  we  see  from  Equation  (76)  that  the  con- 
jugate of  a  scalar  is  the  scalar  itself,  and  that 

Ka  =  —  a  =:  —  TalJa (77), 

or,  the  conjugate  of  a  vector  is  the  vector  reversed.  In  general 
notation  we  may  write 

q  =  ^q  +  Yq, 

whence  it  follows  from  the  above  that 

Kq  =  Hq-Yq  ") 

or  (Art.  43)  [-.....     (78), 

Kg  =  iv  —  xi  —  yj—  zh  ) 

that  is,  the  scalar  of  the  conjugate  of  a  quaternion  is  the  scalar 
of  the  quaternion,  and  the  vector  of  the  conjugate  of  a  quaternion 
is  the  vector  of  the  quaternion  reversed  ;  a  result  which  may  be 
expressed  S3'mbolically 

^K^^=«^^      j (79). 

yKg  =  -Vgj  ^     ^ 

These  ai*e  Equations  (53)  and  (54). 

If  we  add  and  subtract  the  two  conjugate  quaternions 

5'=Sg+Vg,     Kg=Sg— Vg, 

we  have 

g  +  Kg=2Sg| 

g  -  Kg  =  2  Yg  j  ^     ^ 


The  sum  of  tiuo  conjugate  quaternions  is,  therefore,  alioays  a 
scalar,  positive  or  negative  as  the  Z  g  is  acute  or  obtuse.     If 

Z  g  =  -,  this  sum  is  evidently  zero. 

Since,  if  g  is  a  scalar,  Kg  =  g,  then,  conversel}',  ifKq  =  g,  g 
is  a  scalar. 


70 


QUATERNIONS. 


47.   Opposite  Quaternions. 

If,   for  -,  we  write   ^^  (tig-  37),  the  latter  is  called  the 
Opposite  of  g,  and  is  evideutlj-  —  g,  for 


-  =  0  -  7  =  -  g. 


As  appears  from  the  figure,  opposite  quaternions  have  a  com- 
mon plane  and  tensor,  supplementary-  angles  and  opposite  axes ; 
or 

T  (— g)  =  Try,  /.—q  =  tt  —Z.q     and  axis  (—7)  =  —  axis  q. 


Since 


—  a       a  a  —  a 0  /^ 


P  ft  ft  ft 

the  sum  of  two  opposite  qxiaternlons  is  zero^  or 

Fig.  37.  9  +  (-g)  =  0. 


(' 

Also,  since 

—  a      a         —  a 

ft     'ft~     ft 

a 

k:!. ■■    ->-  - 

~~~    ' 

-1, 


or,  their  qxiotlent  \s  negative  nnity. 


If  then 
we  shall  have 


a        Ta  /  ,    ,         •       1  \ 

q  =-  =  —  (cos  (f)  +  €  Sin  <i) 

/i      T/3^        ^  ^^ 

-  7  =  ^ (- COS  «^  -  €  sm<^) 


(81). 


If  Z  7  =  -^,  K7  =  —  g  ;    and,  conversely,  if  Kq  =  —  q,  q  is  a 
vector. 


GEOIVIETRIC    MULTIPLICATION   AND   DIVISION. 


71 


48.  Siuce  Vq  is  independent  of  the  vector  lengths,  and  only 
dependent  npon  relative  direction,  versors  are  equal  whose  axes 
and  angles  are  the  same.     Hence 


UKg  =  U 


But  (Ai-t.  24) 


fij  a        Ua  Uy3 


and,  Equation  (82), 


q       Vq 


(82). 


(83), 


VKq 


Vq 


Again,  since  the  conjugate  of  a  versor  is  the  same  as  the  re- 
ciprocal of  that  versor,  Ave  have,  from  Equations  (82)  and  (83), 


UKg  =  KUg 


(84). 


49.   Representation  of  Versors  by  spherical  arcs. 

If  a,  IS,  y, are  co-initial  unit  vectors,  their  extremities  will 


^  being  any 


all  lie  on  the  surface  of  a  unit  sphere  (Fig.  38) . 

a 
quaternion,  U  ^  turns  (3  from  the  position 

OB  to  OA,  and  this  versor  ma}-  be  repre- 
sented b}'  the  arc  ba  joining  the  vector 
extremities  ;  for  this  arc  determines  the 
plane  of  the  versor  as  also  the  magnitude 
and  direction  of  its  angle,  the  direction 
of  rotation  being  indicated  by  the  order 
of  the  letters  as  in  the  case  of  vectors. 
This  representation  of  versors  b}-  vector 
arcs  is  of  importance  in  the  theox'ems  re- 
lating to  the  multiplication   and  di\ision  of  quaternions,   and 

maj'  be  made  upon  a  unit  sphere  ;  for,  if  a,  fS,  y, are  not  unit 

vectors,  the  quaternions  will  differ  from  the  versors  by  a  nu- 
merical factor  only,  the  introduction  of  which  cannot  affect  the 


72 


QUATEKNIONS. 


versor  conclusions.     Disregarding,  then,  the  tensors,  since  ver- 
so rs  are  equal  whose  planes  are  parallel  anrl  angles  equal  (in- 
cluding   direction),    equal   arcs   on   the 
Fig.  38.  same  great  circle  and  estimated  in  the 

same  direction  represent  equal  vei-sors, 
for  an}'  arc  may  be  slid  OA'cr  the  gi'cat 
circle  on  which  it  lies  without  change  of 
length  or  reversal  of  direction.     On  this 
plan  u'a  =  AB  will  represent  the  recipro- 
cal or  conjugate  of  ba,  and  a  quadrautal 
versor  would  have  for  its  representative 
BC,  an  arc  of  90?     Also,  the  versors  of 
all  compUinar  quaternions  will  be  rei)re- 
sented  bv  arcs  of  the  same  great  cu'cle,  wliile  arcs  of  different 
great  circles  will  represent  the  versors  of  diplanar  quaternions, 
which  are  always  unequal. 

If  M,  N  and  p  are  the  vertices  of  a  spherical  triangle,  the  vector 
arcs  MN,  NP  and  pm  will  represent  versors,  and  it  will  be  seen 
that  by  taking  the  geometric  sum  of  two  of  these  arcs  in  a  cer- 
tain order,  the  remaining  arc  will  represent  the  versor  of  their 
product ;  so  that  if  g'  be  represented  by  pm  and  q  b}'  np,  q'q  may 
be  constructed  b^^  a  process  of  spherical  addition  represented  by 
PM  +  NP  =  KM,  NM  representing  tiie  versor  q'fj  ;  but  that  because 
q'q  and  qq'  are  not  generally  equal,  this  process  of  spherical  ad- 
dition, as  representing  versor  multiplication,  is  not  commutative 
as  was  that  of  vector  addition,  pm  +  np  and  np  -f-  pm  representing 
diplanar  versors. 


50.  Addition  and  Subtraction  of  Quaternions. 

Since  a  quaternion  is  the  sum  of  a  scalar  and  a  vector,  in 
finding  the  sum  or  difference  of  several  quaternions  the  sum  or 
difference  of  their  scalar  and  vector  parts  maj-  be  taken  sepa- 
rately. The  former  will  be  a  scalar  and  the  latter  a  vector; 
consequently,  tJie  sum  or  difference  of  several  quaternions  is  a 
quaternion. 


GEOMETRIC    MULTIPLICATION   AND   DIVISION.  73 

1.  Both  the  associative  and  commutative  priuciples  being 
apphcable  to  the  summation  of  scalars,  as  also  to  that  of  vectors 
(Arts.  4,  5),  the}-  also  hold  good  for  the  addition  and  subtrac- 
tion of  quaternions ;  or 

q  +  r  =  r-{-q  ~) 

and  [    .     .     .     .    (85). 

q  +  {r+s)^{q  +  r)  +  s) 
If  then 

q  =  Hq+yq 
r  =  Hr  +V)' 

s=q+r-{- =  Ss+Ys; 

in  which 

Ss  =  S  (q  +  r  + )  ^  Sq  +  8r  + , 

Ys  =  Y(2  +  r  + )  =  Yg  +  Yr  + , 


and,  in  general, 


^Iq ^  ^^q  \ 
T:Sq  =  %Yq  J 


(SG), 


or,  in  quaternion  addition  and  snhtraction,  S  and  Y  are  distribic- 
tive  symbols. 

2.  If  f/  +  r  +J7  -f- =  s,  then,  Equation  (78) , 

Kg+  Kr  +  E^)  + =  Sq+  Sr  +  Sp  + -Yq-\r--Tp  - 

=  Ss  —  Ys  =  Ks. 

.-.   lKq  =  K^q (87), 

K,  like  S  and  Y,  being  a  distributive  s^-mbol. 

3.  Again,  since  the  conjugate  of  a  scalar  is  the  scalar  itself, 

KS^  =  Sq. 
But  Sg  =  SKg.     Hence 

KS5  =  Sr^  =  SKg (88). 

Also,  since  the  conjugate  of  a  vector  is  the  vector  reversed, 

K\q  =  -\q. 


QUATERNIONS, 


But  —  yq  =  VKg.     Hence 
hence  K  is  commutative  loith  S  and  Y. 


(89); 


Fig.  38. 


4.  Since  an}'  two  quaternions  ma}'  be 
reduced  to  a  common  denominator  (Art. 
44),  so  that 


and  since 


a       y_a4-y_ 


Ta'+Ty'>T(a'+y) 

unless  a'  =  .Ty'  anda;>0,  it  follows  that 

1q  +  Tq'>'i{q  +  q') 

unless  q-=xq'  and.r>0.  Hence,  in  general,  T2g  is  not  equal 
to  2Tg.  Moreover,  since  U2g  is  a  function  of  the  tensors  under 
the  2  sign,  while  2U7  is  independent  of  the  tensors,  Vlq  is  not 
equal  to  2U<^.  This  also  appears  from  the  representation  of  ver- 
sors  by  spherical  arcs  (Fig.  38).  Hence,  in  the  addition  and 
subtraction  of  quaternions,  T  and  U  o?'e  not,  in  general,  dis- 
tributive symbols. 

51.  Multiplication  of  Quaternions. 

1-    Let 

q  =  Sq+^q,     r=hr+^r 

be  any  two  quaternions.     Then 

p  =  qr  =  SryS/-  +  Sq\r  +  SrVry  +\qYr. 

The  last  member,  being  the  sum  of  a  scalar  and  a  vector,  is  a 
quaternion.  Hence,  the  product  of  two  quaternions  is  a  quater- 
nion, and 

p  =  SjJ  +Vi)  =  ^qr  +yqr, 
in  which 

Sgr  =  S^Sr  +  S  .  YvVr       ....     (90), 
and 

\qr  =  ^q\r  +  ^ryq+y  .\qyr     .     .     .     (91). 


GEOMETRIC   MTJLTIPLTCATION   AND   DIVISION.  75 

If  we  multiply  ^.b}'  r,  Ave  obtain 

Srg  =  SrSg  +  S  .  YrYq, 

Yrq  =  Hr\q  +  SfyVr  +V  .  YrYq. 

But,  Equation  (53), 

S  .  YrYq  =  S  .  YqYr. 
...   Srq  =  fiqr (92). 

But,  Equation  (54), 

Y.YqYr  =  -Y  .YrYq, 


and  therefore  the  products  qr  and  rq  are  not  equal.  Hence, 
quaternion  multi'plication  is  not  in  general  commutative.  If, 
howGA'er,  q  and  r  are  complanar,  Yq  and  Yr  are  parallel,  and 
V  .  YqYr  =  0  ;  in  which  case  qr  =  rq.  Conversely,  if  qr  —  rq,  q 
and  r  are  complanar. 

Since  Reciprocal,   Conjugate   and  Opposite  quaternions   are 
complanar,  they  are  commutative,  or 


qKq  =  Kq  ,  q 
q  -  =  -q  =  qq-^  =  q-^q 


.     (93). 


2.    It  has  been  shown  (Art.  44)  that  any  two  quaternions 

q,  q',  can  be  reduced  to  the  forms  "  and  Z  having  a  common 

a  a 

denominator,  or  to  the  forms  "  and  1.     Hence 

o  a 


"We  have  then 


,  y      /S       y      a       y 

q'  '.q  =  l:^  =  L         =1 

a      a       a     [5      p 


g  /8       T^       Ta      T^       Ta      Ta  ^   *      ^ 

ry         (3      VI3      Ua     U/3      Ua'Ucc      ^'^'^"i 


(94). 


76  QTJATEBNIONS. 

In  a  similar  mauucr 


^        -•  •  (95). 


V{r/q)  =  uI=Vq'\]q 
6 

Hence  the  tensor  of  the  product  (or  quotient)  of  any  two  qua- 
ternioiis  is  the  j)roduct  (or  quotient)  of  their  tensors,  and  the  ver- 
sor  of  the  j)roduct  (or  quotient)  is  the  x)roduct  (or  quotient)  of 
their  versors. 

In  foct,  tensors  being  commutative,  we  have,  in  general, 

Tng  =  nT(7 (96), 

nry  =  TUq  .  Vliq  =  HTg  .  liVq, 

.'.  Unry  =  nUg (97). 

3.  The  multiplication  and  division  of  tensors  being  purely 
arithmetical  operations,  we  proceed  to  the  cori'esponding  opera- 
tions on  the  versors.  It  has  been  shown  (Art.  41)  that  an}'^ 
two  versors  5,  q',  may  be  reduced  to  the  forms 

B      on         ,      y'      oc'  /T-,.      on\ 

A,  B,  c',  being  the  vertices  of  a  spherical  triangle  on  a  unit 
sphere.     Then 

,A  =  t.^  =  l'  =  2£'. 

fS       a        a       OX 

If  we  represent  the  versors  q'  and  q  liy  the  vector  arcs  i;c' 

and  An,  then  the  versor  ^,  the  product  of  q'q,  will  be  rcprc- 

«  y' 

sented  b}-  the  arc  ac'  ;  moreover  if  q"  =  -   represent  anv  divi- 

(3  .  a  - 

dcnd  and  q=  -  an}-  divisor,  then 

q  ^  a  '  (3^ ^~  oa* 
the  versor  of  the  product  q'q  being 

BC'  +  All  =  AC', 


GEOMETRIC   MULTIPLICATION   AXD   DIVISION. 


77 


Fi-.  39. 


aud  the  versor  of  the  quotient  i_ 

AC'— AB  =  BC'; 

and,  as  in  the  addition  aud  subtraction  of  quatcruious,  the  pro- 
cess consisted  in  an  algebraic  addition  and  subtraction  of  scalars 
but  a  geometric  addition  and  sul)trac- 
tion  of  vectors,  so  the  multiphcation 
and  division  of  quaternions  is  reduced 
to  the  corresponding  arithmetical  ope- 
rations on  the  tensors  and  the  geome- 
trical multiplication  and  division  of 
the  versors,  the  latter  being  con- 
structed by  means  of  representative 
arcs  and  the  rules  of  spherical  addition  and  subtraction. 

4.  The  representation  of  a  versor  by  the  arc  of  a  great  circle 
on  a  unit  sphere  illustrates  the  non-commutative  character  of 
quaternion  multiplication.  For,  ab  and  ba'  (Fig.  39)  being  equal 
arcs  on  the  same  great  circle,  as  versors 


and  similarly 
Now  if 

then 


AB  =  BA 


CB  =  BC 


g  = 


a'/3 


li 


and 


y 


=  —     and 

7 


the  versors  qr  and  rq  being  represented  b}-  the  arcs  ca'  and  ac' 
respectively.  These  arcs,  though  equal  in  length,  are  not  in  the 
same  plane,  and  therefore  the  versors  rq  aud  qr  are  not  equal. 
Constructing  these  versors,  by  spherical  addition  we  should  have 

BC'  +  AB  =  AC', 
AB  +  BC'  =  ba'  +  CB  =  ca', 

a  change  in  the  oi'der  giving  unequal  results. 


78  QUATERNIONS. 

Hence,  unless  ac'  and  c.v'  lie  on  the  same  gi'cat  circle,  In 
which  case  q  and  r  are  complanar,  quaternion  multiplication  is 
not  commutative. 

5.  Other  results,  hereafter  to  he  ohtained  symliolically,  may 
be  readily  proved  by  means  of  spherical  arcs,  as  follows  : 

If  AB  (Fig.  30)  represents  the  versor  of  g  =  -,  a  b  =  ba  repre- 

1  "^ 

sents  the  versor  of  Yiq  or  -.     The  spherical  sum  of  ab  +  ba 

'^  1 

being  zero,  the  effect  of  the  versors  in  the  products  qKq  and  q- 

is  to  annul  each  other.     Hence,   if  the  vectors  are   not  unit 
vectors,  "  qKq  =  ILq  .  q  =  {TqY (98), 


Again,  from 

,1=1, =1. 

ab  -f  BC'  =  Ca', 

we  have 

qr  =  -, 

y 

and  the  versor  of  K  {qr)  will  therefore  be  represented  b}-  a'c. 
But 

a'c  =  BC  +  a'b, 

whence 

lL{qr)  =  JLrlLq (99), 

or,  the  conjugate  of  the  product  of  tioo  quaternions  is  the  irroduct 
of  their  conjugates  in  inverted  order. 

6.  The  product  or  quotient  of  complanar  quaternions  is  readilv 
derived  from  the  foregoing  explanation  of  versor  products  and 
quotients  as  dependent  uj^on  a  geometric  composition  of  rota- 
tions. For,  disregarding  the  tensors,  the  vector  arcs  which 
represent  the  versors,  since  the  latter  are  complanar,  will  lie  on 
the  same  great  circle,  and  the  processes  which  for  diplanar  ver- 

sors  were  geometric  now  become  algebraic.     Thus  for  </  — 7> 

a  qq'=q'q=       ^'d=-, 

p      a       a 


GEOMETEIC   MULTIPLICATIO:^:   AND   DIVISIOX.  79 

and,  Fig.  39, 

ba'  +  ab  =  ab  +  ba'  =  aa'  ; 

also  for  (/"=  -  aud  fj'=  -, 


and 


^_^.^_^      a_a' 
q'        a  *   a        a      /3       jS' 

BA  +  aa'  =  ba'. 


Fis.  39. 


The  product  or  quotient  of  an}-  two  complauar  quaternions  is 
therefore  obtained  by  multiplying  or 
dividing  their  tensors  and  adding  or 
subtracting  their  angles.     Thus 


2jq  =  Ti)  .  Tq  [cos  (cf> -{- 0) -{- 
esin(c^  +  ^)]. 
If  J9  =  q, 


f/  =  {Tqf  (cos  2  </)  +  e  sin  2  <^) , 

or,  generally, 

5"=(T(7)"(cosn^  4-esin?i^) 

whence  result  the  following  general  foimulae, 

u(r)  =  (x:g)" 

Sl](q")  =  cosnZq 
TYU  (q")  =  sin  n  Z  q  _ 

which  are  all  involved  in  Art.  42. 


(100), 


(101), 


52.     1.    Distributive   and  Associative  Laws   iii  Vector 
and  Quaternion  Midtiplication. 

Having  assumed  (Art.  24) 

-  +  -= 1 

a       a  a 


80  QUATERNIONS, 

whence 

since  a  is  anj-  vector,  we  have 

^a  +  ya  =  (^  +  y)a.  (a) 

Taking  the  conjugate  of  (/?  +  7)0, 

K[(/5  +  7)a]  =  KaK(/?-|-y)  [Eq.  90] 

=  Ka(Ky3  +  Ky).  [Eq.  87] 

Taking  the  conjugate  of  {(3a  +  ya) , 

K(^a  +  ya)  =  K^a  +  Kya  =  KaK/3  +  KaKy. 
Hence 

Ka(K/?+  Ky)  =  KaK/3  +  KaKy, 

or 

a'(/3'+y')=a'^'+aV.  (b) 

Hence,  from  (a)  and  (5),  the  multiplication  of  vectors  is  a 
doubly  distributive  operation^  and 

(/?+7)('^  +  S)  =  ;8a  +  ya  +  /38  +  y8      .      .      (102). 

2.    Let  g  =  ~-,  he  any  quaternion  and  a  any  vector ;  also  /3  a 

vector  along  the  line  of  intersection  of  a  plane  ])eqiendicular  to 
a  with  the  plane  of  cj.  Then  another  vector,  8,  niaj-  be  found  in 
the  latter  plane,  such  that  q=^i  C  ha^^ng  the  same  angle,  plane 
and  axis  as  '—.  Also  let  y  be  a  vector  in  the  intersecting  plane, 
such  that  -^  =  a.     If  now  a  be  any  scalar, 

^  a/3  +  y     /3^af3  +  y 
~      13      '  S  8 

=  ag  +  ag. 


GEOMETRIC   MULTIPLICATION  AND   DIVISION.  81 

Taking  the  conjugates  as  above, 

q'{a'+a')  =  q'a'+q'a'. 

Hence,  in  general, 

(a  +  a)  («'+  a')  =  aa'-\-  aa'+  a'a  +  aa';  (c) 

or  regarding  o,  a',  and  a,  a'  each  as  the  sum  of  two  scalars  and 
two  rectors  respective!}', 

(«i+  «2  +  «!  +  02)  («'i  +  a'2  +  a'l  +  a'2)  = 

(oi  +  Oo)  (a'l  +  a'o)  +  (ai  +  a.^  (a\  +  a'.)  +  (a'^  +  a',) 

(«!+  02)  +  («!  +  ao)  (a'l  +  a'o)  = 

(«i  +  ai)  (a'l  +  a'l)  +  (f'l  +  ai)  («'2  +  a'o)  +  (Oo  +  a2) 
(«'i  +  a'i)  +  (ao+ao)  (a'o  +  a'2), 

since,  from  (c),  the  factors  in  the  expression  preceding  the  last 

are  distributive.     Putting  for  the  parentheses,  which  are  sums 

of  a  scalar  and  a  vector,  the  quaternion  sjTubols  p,  q,  r  and  s, 

we  have 

{p  +  q){i'  +  s)=pr+ps  +  qr  +  qs    .     .     (103), 

or,   tJie   multiplication   of  quaternions   is   a   doubly  distrihutive 
operation. 

3.    Assuming  any  three  quaternions  under  the  quadiinomial  ' 
form,  Article  43,  i,  k,j  being  unit  vectors  along  three  mutually 
rectangular  axes,  we  have 

q  =  iu  -\-xi  +yj  +zJc,  (a) 

r  =  iv'  +  x'i  +  y'j  +  z'Jc,  (&) 

s  =  ^y"+  x"i+  y"j+  z"k.  (c) 

MultipMng  first  (c)  by  (6)  and  the  result  by  (a),  and  then 
(6)  \)j  (a)  and  (c)  by  this  result,  observing  the  order  of  the  fac- 
tors, it  will  be  found  that  the  scalar  and  vector  parts  of  these 
two  products  are  respectivel}-  equal,  and  therefore 

q{rs)=^{qr)s (104), 

or,  the  associative  law  is  true  in  the  multiplication  of  quaternions. 


82  QUATERNIONS. 

53.     1.    If  a  and  /?  be  au}-  two  vectors,  then 

(a  +  /3){a  +  f3)  =  (a+  fSy  =  a'  +(a(3  +  (3a)  +  /^\ 

whence,  Equation   (55),  or,  comparing  Equations  (39),   (41) 
and  (80), 

(a+/3)-  =  a2  +  2Sa/?  +  ^2        _       _       ^      (^^Q^y 

2.    Similar! V 


or 


(^_/J)(a_^)  =  (a-^)2  =  „2_(„^^.^a)+/3^ 

(a-^)-  =  a--2Sa/3  +  /3-        .       .       .      (IOC), 


3.    From  Equation  (57),  or  by  multiplying  q  =  iiq  -\-\q  iuto 

aft,fSa  =  (Sqy--{\qy-; 

hence,  from  Equation  (08),  the  equalities 

a^  .  (3a  =  qKq  =  {SafSy--{\afty  =  (Tqy      .      (107). 

54.  Applications. 

1.    Ill  any  right-anc/Ied  triangle,  the  square  on  the  hypothenuse 
is  equal  to  the  sum  of  the  squares  on  the  sides. 

Let  the  sides,  as  vectors,  be  repre- 
^'^■^-  sented  by  a  and  /8  (Fig.  40),  and  the 

hypothenuse  by  y.     Then 


or,  Alt.  41,  4, 


Squaring,  Equation  (105), 
y-'=a2+2Sa;8  +  y3^ 
/  =  a2  +  /3^ 
or,  as  lengths  simplv,  changing  signs  [Equation  (33)], 

BA-  =  BC^  +  CA^. 

2.    In  any  right-angled  triangle,  the  medial  to  the  hypothenuse 
is  one-half  the  hypothenuse. 


GEOMETRIC   MULTIPLICATION   AND   DIVISION.  83 

111  Fig.  40,  for  the  medial  vector  cd  =  S,  we  liave  (Art.  15) 

or 

2S  =  /3-a. 

Squaring,  and  since  S,3a=  0, 

4S-"  =  /5-  +  aS 
or 

CA-  +  CB-         Ab' 


CD 


4  4 

AB 


3.  If  the  diagonals  of  a  parallelogram  are  at  rigid  angles  to 
each  other,  it  is  a  rhombus. 

Let  tlie  vector  sides  be  represented  by  a  and  /3-     Then  a  +  yS 
and  a  —  /5  are  the  vector  diagonals. 
By  condition 

S(a  +  /5)(a-^)  =  0.  [Art.  41,  4] 

But,  Equation  (53), 

S(a  +  /?)  (a  -  /5)  =  a^  -  /3-  =  0, 

which  is  true  only  when  Ta  =  T/3,  that  is  when  the   sides  are 
equal. 

4.  The  figure  formed  by  joining  the  middle  p)oints  of  the  sides 
of  a  square  is  itself  a  square. 

Let  BC  and  ca  (Fig.  40)  be  the  sides  of  a  square,  p  and  q 
their  middle  points,  and  o  the  middle  point  of  the  side  opposite 
BC.     Then,  with  the  same  notation, 

PQ  =  i(a  +  /3),  QO  =  i(/3-a); 

.-.    S(PQ  .  QO)  =  0, 

or  PQ  and  QO  are  at  right  angles. 

5.  In  any  triangle.,  the  square  of  a  side  opposite  an  acute 
angle  is  equal  to  the  sum  of  the  squares  of  the  other  sides,  less 


84 


QUATERNIONS. 


tivice  the  product  of  the  base  and  the  line  between  the  acute  angle 
and  the  foot  of  a  perpendicular  from  the  angle  opposite  the  base. 


Fig.  41. 


or 


Let  CA  =  f3,  CB  =  a,  BA  =  y  (Fig.  41). 
Then 

/3  =  a  +  y, 
y3=^  =  a2  +  2Say  +  /. 

Now 

2Say  =  -  2TaTy  COS  (180°-  li)  = 
20CC0SB. 

Heuce 


—  h-  =  —  a-  —  cr  -\-2ac  cos  n  =  —  a-  —  cr  -\-  2  ad, 
b-  =  a-  +  r  —  2  ad. 
If  B  is  a  right  angle,  Say  =  0,  and,  as  in  Example  1, 

b-  =  a-  +  (.". 


What  does  this  theorem  become  for  a  side  opposite  an  obtuse 
angle  ? 

6.    In  any  j)lane  triangle,  to  find  a  side  in  terms  of  the  other 
tico  sides  and  their  opposite  angles. 
In  Fig.  41, 

^=a  +  y. 

Multiplying  into  a 

fSa  =  a"  +  ya. 

Taking  the  scalars  (Art.  41,  "j), 

S,5a  =  -  cr  +  Sya, 
or 

—  ba  cose  =  —  a-  —  ca  cos  (180°—  b)  ; 
,'.  a  =  b  cose  +  c  cosB. 

The  above  operation  with  a  is  indicated  b}'  sa3'ing  simplj', 
"  operating  with  X  S  .  a,"  meaning  that  a  is  first  introduced  and 
then  the  scalars  taken.     The  position  of  the  sign  X  will  indicate 


GEOINIETRIC   MULTIPLICATION   AND   DIVISION. 


85 


hoiv  a  is  used.     K  used  as  a  multiplier,  we  should  write,  "  oper- 
ating; with  S  .  a  X  ." 


7.  The  sines  of  the  angles,  in  any  plane  triangle,  are  propor- 
tional to  the  opposite  sides. 

In  Fig.  41 

Operating  witli  X  V .  a,  that  is,  as  explained  in  the  preceding 
example,  multiplying  into  a  and  taking  the  vectors  (Ail.  41,  5), 

Ty8a  =T(a  +  y)  a  =  V  ,  a"  +Vya. 

But  V .  a^  =  0  ;  hence 

T/?a=Tya, 
ha  sine  =  ca  siuB, 
or 

sine  :  siuB  :  :  c  :  &. 

Notice  that  V/?a  and  Vya  involve  a  unit  vector  at  right  angles 
to  their  plane,  and  that,  owing  to  the  order  of  the  vector  factors, 
€  has  the  same  sign  in  both  members  of  the  equalit}',  and  may 
therefore  be  cancelled.  The  period  in  V .  a?  ma}'  evidentl}'  be 
omitted,  as  in  Y^a ;  it  will  be  used  hereafter  only  to  avoid  am- 
biguity. Thus  Kgr  means  the  conjugate  of  qr ;  but  Kg  .  ?•  is  r 
multiplied  by  the  conjugate  of  g. 

8.  In  a  right-angled  triangle,  to  find  the  sine  and  cosine  of  the 
acute  angles. 

Let  AB  =  y,  AC  =  ^,  BC  =  a  (Fig.  42) . 

Then 

/8  =  y  +  a, 
whence 

Taking  the  scalars,  since  S  — =  0, 


Fig.  42. 


l=rCOSA,     or     COSA=-- 
b  c 


86 


Fig 

42. 

A 

c 
/ 

a 

QUATERNIONS. 
Takius:  the  vectors 


^■^^■^«' 


c    . 

-  sill  A 

6 


sine  =  0  ; 


sin  A  =  -  • 
c 


In  this  example  TH''^  =  —  UV  ^• 

9.    To  find  the  sine  and  cosine  of  the  sum  of  two  angles. 
Let  a,  /?,  y  1)0  coinplanar  unit  vectors  (Fig.  43),  and  e  a  unit 
vector  perpendicular  to  theh'  plane.     "We  have 


Fi-.  43. 


in  which 


^  =  cos(</)  +  ^)  +  £  sin(</)  +  ^) , 

y 


y3 


=  cos  <i  +  c  Sin  ( 


-  =  cos  6  +  e  sin  ( 


Hence 


cos{(f)  +  0)  +  esin(^  +  0)  =  (cos(j>  +  esin<^)  (cos^  +  csin^) 
=  cos^  cos 6  +  e(sin</)  cos^  +  cos^  sin^)  +  €-sin^  sine/). 

Equating  the  scalar  and  vector  parts  in  succession,  there  re- 
sults, since  e^  =  —  1 , 

cos (cf>-\-6)=  cos <f>  cos ^  —  sin  ^  sin 6, 
sin  {<f>  -\-  6)  =  s'm  (fi  cosO  +  cos0  sin^. 

10.    To  find,  the  sine  and  cosine  of  the  difference  of  two  angles. 
Let  the  angle  between  y  and  a  (Fig.  43)  be  ij/.     Then 

y       a     y' 


GEOMETRIC   IMTJLTIPLICATIOJf   AXD   DI\^ISION. 
in  which 


87 


cos(i//  —  6)  —  esiu(t/^  —  0), 


y' 

-  =  cosO  +  esin^, 
a 

_  =  COS  ij/  —  e  sin  ij/, 


and,  as  in  the  preceding  example, 

cos(i/^  —  d)z=  cosO  cost//  +  sin^  sini/', 
sin  ({j/  —  0)  =  cos  6  sivnj/  —  sin  d  cos  i{/. 

11.  If  a  straight  line  intersect  txvo  other  straight  lines  so  as  to 
make  the  alternate  angles  equal,  the  two  lines  are  parallel. 

Let  a  and  y  (Fig.  44)  be  unit  vectors  along  ab  and  cd,  and 
yS  a  nnit  vector  along  ac.     Then 


whence 


af3  =  —  cos  6  -\-  e  sin  $, 
Py  =  —  cos  6  —  i.  sin  9  ; 

a.p  -/3y=2  Va/3, 


Fig.  44. 


and  therefore,  Eqnation  (56),  y  =  a. 
If  a  =  AB,'  then 

a(3  =  cos  (9  —  €sin^, 
(By  =  —  cosO  —  csin^, 
a/3-(3y=2^a(3; 

•■•  y  =  — a- 


[Eq.  (55)] 


12.  If  a  parallelogram  he  described  on  the  diagoncds  of  any 
p>arallelogram,  the  area  of  the  former  is  twice  that  of  the 
latter. 

Let  a  and  /?  represent  the  sides  as  vectors  ;  then  the  diagonals 
are  a  +  ^  and  a  —  (3,  and 

V(a  +  ^)  (a  -  (3)  =Y(/3a  -  a/S)  =  2Y/3a, 


since  Ya^=Y/3"-  =  0  and  -Va^  =  V/3a. 


88  QUATERNIONS.  / 

But,  from  the  order  of  the  factors, 

UV(a  +  /3)(a-^)=UV)8a, 

hence 

TV(a  +  ^)(a-/?)  =  2Tyy3a, 

which  is  the  proposition  (Art.  41,  7). 

13.    Par allelocj rams  on  the  same  base  and  between  the  same 

parallels  are  equal. 

We  have  (Fig.  45) 


Fig.  45. 


Fip:.  46. 


BE  =  BA  +  AE 
=  BA  +  .KBC. 

Operating  with  V .  bc  X 

V(bc  .  be)  =  V(bc  .  ba), 
since  Va'BC-=  0. 


BC  .  BE  sin  EBC  =  BC  .  BA  Sin  ABC, 

which  is  also  true  when  tlie  bases  are  equal,  l)ut  not  co-incident. 

14.  If ,  from  any  point  in  the  j)/a«e  of  a  parallelogram^  per- 
peiidiculars  are  let  fall  on  the  diag- 
onal and  the  tico  sides  that  contain 
it,  the  product  of  the  diagonal  and 
its  j^^n^c^dicular  is  equal  to  the 
su7n,  or  difference,  of  the  products 
of  the  sides  and  their  respective  per- 
pendicidars,  as  the  j)oint  lies  loith- 
out  or  within  the  parallelogram. 

Let  OA  =  a,  OB  =  /?,  OP  =  p  (Fig.  4G) . 
Then 

Vap+V/3p=V(a  +  /3)p. 

But 

UYap  =  UV/3/3=  UV  (a  +  p)p. 
Hence 

TVap  +  TV/?p  =  TV(a  +  p)p. 


GEOMETEIC    MULTIPLICATION   AND   DIVISION.  89 

For  p'  =  op',  we  have 

V\ap'=  -  VY(3p'=  ±  UY(a  +  /?)p'; 
.-.   TYap'^T\(3p'=      TY{a  +  ft)p: 

15.  If,  on  any  two  sides  of  a  iriangle,  as  ac,  ab  (Fig.  47). 
any  two  exterior  parallelograms,  as  acfg,  abde,  he  constructed, 
and  the  sides  ed,  gf,  produced  to  meet  in  ir,  then  ivill  the  sum  of 
the  areas  of  the  parcdlelograms  he  equal  to  that  ivhose  sides  are 
equal  and  parallel  to  cb  and  ah. 

Let  AE  =  a,  AB  =  /?,  AC  =  y 

and  AG  =  8.  Then 

AH  =  AE  +  EH 
=  a  —  X[B. 


Operating  with  x  V  .  /? 

V(AH.^)=ya;S. 

We  have  also 
Operating  with  x  V .  y 


(a) 


AH  =  AG  +  GH 

=  8  -  yy. 


V(AH.y)=V8y.  (&) 

Hence,  from  (a)  and  (&), 

VAH(/3-y)=Va/?-V8y, 
Y(aH  .  CB)=Va/3-VSy=Tay3+Yy8. 

These  vectors  have  a  common  versor ;  whence  the  proposition. 
If  one  of  the  parallelograms,  as  ad',  be  interior,  then  ae'=  —  a 
and  ah'  =  —  a  —  x'^  =  8  +  y'y,  and 

V(AH'./3)  =  -Va/3, 

V(An'.y)=V8y; 
.-.      VAn'(/3-y)  =  -Va^-VSy=Y/?a-VSy. 


90  QUATERNIONS. 

But  in  this  case 

rV(AH' .  cr)  =  -  UT/?a  =  -  UTSy, 

and  the  area  of  the  parallelogram  on  aii',  cb,  is  the  area  of  af 
minus  the  area  of  ad'. 

1 G.    To  find  the  angle  behoeen  the  diagonals  of  a  parallelogram. 

Let    AD  =  Bc  =  a    (Fig.    48) , 
Fig.  48.  and  BA  =  CD  =  /?,  d  and  d'  being 

the    tensors    of   the   diagonals. 
Tlien 

AC  .  DU  =  -  (a  -  ^)  (a  +  ^) 

=  -a?-{a(i-Pa)  +  l? 
=  -  a-  -  2  Va;8  +  ;8-. 

Taking  the  sealars 

cos  DOC  .  dd'=  a-  —  b-. 
Taking  the  vectors 

sin  DOC  .  dd'=2absm0, 

since  UV(ac  .  db)  =  -  UVa^. 

.    +          ^          4.      ^      2o&sin^ 
.'.  tan  doc  =  —  tan  ^  =  — ; —. 

or  —  b- 

17.  TJie  sum  of  the  sqxiares  on  the  diagonals  of  a  parallelo- 
gram equals  the  sum  of  the  squares  on  the  sides. 

In  Fig.  48 

BD^  =  (a  +(3)-=  a' +  2  Saft  +(i\ 
ca2  =  (y8  -  a)-  =  /3-  -  2  Sa/3  +  a?  ; 

.-.    CA2+BD-=2a-  +  2/32, 

or 

BD^  +  CA-  =  BA-  +  AD-  +  DC"  +  CB". 

18.  The  sum  of  the  squares  of  the  diagonals  of  any  quadri- 
lateral is  twice  the  sum  of  the  squares  of  the  lines  joining  the 
middle  points  of  the  opposite  sides. 


GEOMETEIC   MULTIPLICATION   AND   DIVISION. 


91 


Let  AB  =  a,  AD  =  /3,  DC  =  y  (Fig.  49).     For  the  squares  of 
the  diagonals,  we  have 


Fig.  49. 


(/?  +  y)-"  +  (/3-a)^ 
and  for  the  bisecting  hues 

Whence  the  proposition  readil}'  follows. 


19.  The  sum  of  the  squares  of  the  sides  of  any  quadrilateral 
exceeds  the  sum  of  the  squares  on  the  diagonals  by  four  times  the 
square  of  the  line  joining  the  middle  points  of  the  diagonals. 

Let    AB  =  a,     AC  =  /?,     AD  =  y 

(Fig.   50).      The  squares  of  the 
sides  as  vectors  are 


or 


a^  +  (/3-a)-^  +  (y-/3)^  +  yS 
or 

2(a2  +  /3-  +  y2)  -  2  S^a  -  2  Sy/?. 
The  squares  of  the  diagonals  are 

^^  +  (y-a)^ 
/32  +  y2  +  a-  -  2  Sya. 

The  former  sum  exceeds  the  latter  by 

aH /S^  +  y- -  2  S/3a  -  2  Sy^  +  2  Sya, 

(a-i8  +  y)S 
which  may  be  put  under  the  form 

a  +  y       (SV 


or  by 


9-2 


QFATERNIONS. 


But  "lJU  =  AG,  and   —  ''  =  s>a 
2  2 

we  obtain 


Substituting  these  values, 


4(ao  +  sa)-,  or  4so^, 
wliich  is  also  true  of  the  vector  lengths. 

20.    In  any  quadrilateral^  if  the  lines  joining  the  middle  points 
of  opj)osite  sides  are  at  right  angles,  the  diagonals  are  equal. 
With  the  notation  of  Fig.  49,  we  have 

FK.GII=[H7-a)+/?]Ka+y). 
But,  by  condition, 

S(FE  .  OIl).=  }^  _  ^V  ^  +  ^  =  0. 

^  ^44^2^2 


Whence 


FiL'.  51. 


or 


or 


AC-  =  BD^, 
AC  =  BD. 

21.  In  any  quadrilateral  jirism.  the 
sum  of  the  squares  of  the  edges  exceeds 
the  sum  of  the  squares  of  the  diagonals 
by  eight  times  the  square  of  the  line 
joining  the  points  of  intersection  of  the 
two  2xdrs  of  diagonals. 

Let  OA  =  a,  OB  =  /?,  oc  =  y,  CD  =  S 
(Fig.  51).  For  the  sura  of  the 
squares  of  the  edges  we  have 


2[a-"  + /5- +  (8 -a)-+2/+(S -/?)=], 

2[2 a-  +  2^-  4-  2 /  +  2 82  _  2  S8a  -  2  S^^].  (a) 

The  sum  of  the  squares  of  the  diagonals  is 

{y-h^y  +  {y-8yi-{y  +  a-(3y-  +  (y  +  (3-ay, 

2(a-  +  ^=  +  S-  +  2/-2Sa^).  (6) 


GEOMETKIC   MULTIPLICATION   AND   DIVISION,  93 

The  vectors  to  the  intersections  of  the  diagonals  are 
U^  +  y)     and     |(y  +  a  +  ^), 
and  the  vector  joining  these  points  is 

Squaring  and  multiplying  by  eight,  we  have 

2la^-\-(3-  +  S-  +  2 Sa/?  -  2  SaS  -  2 S^8], 
which  added  to  (6)  gives  (a) . 

22.  In  any  tetraedron,  if  two  pairs  of  opposite  edges  are  at 
right  angles,  respectively,  the  third  pair  will  be  at  right  angles. 

Let  OA  =  a,  OB  =  /?,  oc  =  7  (Fig.  52) . 
The  conditions  give 

Sa(^-y)  =  0, 
S/3(a-y)  =  0. 

Subtracting  the  first  of  these  equa- 
tions from  the  second 

Sy(a-^)  =  0, 

which  is  the  proposition. 

23.  To  find  the  relations  betiveen  the  edges,  plane  angles  and 
areas  of  a  tetraedron. 

With  the  notation  of  Fig.  52,  we  have 


or 


CA  .  CB=(a  — y)(^  — y), 
CA  .  CB  =  a/3  —  ay  —  yy8  +  y^. 


(a) 


Representing  the  tensors  of  ca  and  cb  by  m  and  n,  and  taking 
the  scalars  of  (a) , 

S(CA  .  cb)  =  Sa/3  -  Say  -  Sy^  +  y^ 
whence 

(?  —  ac  cos  Aoc  —  he  cos  boc  =  mn  cos  acb  —  ah  cos  aob, 


94 


QUATERNIONS. 


which  is  the  relation   between   the   edges   and   their   included 
angles. 

Taking  the  vectors  of  (a),  and  squaring, 

[y(CA  .  en)]'  =  (Va/S)-  -ya/3Vuy  -Ya^Vy/S  -\ay\a/3  \     .j, 

+  (Yay)'-'  +  VayVy/g  - \y(3\a/3  -f  Vy/3Vay  +  (Vy/?)^  i     ^ "'' 


But 


{\aft\yp  +\yftYa(3)  =  -  2  S  .  Va/3Vy^      (Eq.  55) 

=  2TVn/3TVy/?  COSB, 


in  which  b  is  the  angle  between  the  planes  aou,  boc. 
Also 

-  (VaySVay  +VayVa/3)  =  -  2  S  .  Va/3Vay  =  2  TYa/STYay  COS  A, 

and 

YayYy/8  +  Vy/SYay  =  2  S  .  VayTy^  =  -  2  TVayTVyyScos  (180°-  c) 

=  2TVayTVy/3cosc, 

in  which  a,  b  and  c  are  the  angles 
opposite  the  edges  bc,  ac  and  ab  re- 
spectively.    Hence  (b)  becomes 

-  [TV(CA  .  Cb)]-=  -  (TYafSy--  (Tyay)2 

-iT\y(3f 

+  2  TYa/S  T Vay  COS  A  +  2  TYa/STYy/B  cos  B 

+  2TVayTVy;8c0SC. 

But  (Art.  41,  7th) 

TV(cA  .  cb)  =  2  area  acb, 

and  similarl}-  for  the  others.     Hence,  dividing  by  —4, 

(areaABc)^=  (area  aob)-  +  (ai-ea  aoc)-  +  (area  boc)-  — 

2  area  aob  area  aoc  cos  a  —  2  area  aob  area  boc  cos  b  — 
2  area  aoc  area  boc  cose. 


which  is  the  relation  between  the  plane  faces  and  their  included 
angles. 


GEOMETRIC   IVnrLTIPLICATIOX   AXD   DIVISION. 
If  the  angles  are  right  angles,  then 

(area  ABc)-  =  (area  aob)^  +  (area  aoc)^  +  (areaBOc)- 


95 


24.    To  inscribe  a  circle  in  a  given  triangle. 

Let  a,  p,  y  (Fig.  53)  be  unit  vec- 
tors along  the  sides.  Then,  Ait.  16, 
the  angle-bisectors  are 

-  2/  (y  +  ") , 

Now 

«^(^  +  7)  =  cy-2/(y  +  a), 

Operating  with  V  .  (y  +  a)  x 

^  -         <^^«y 

Vy^+Va/3+Vay' 

Hence 

A0  =  .T(/3  +  y)  = '^^^ 

or,  since  a,  /?,  y  are  unit  vectors, 
c  sin  B 


Fig.  53. 


(/5  +  y), 


AG  = 


sin  A  4-  sin  b  +  sin  c 


(/S  +  y). 


Squaring,  to  find  the  length  of  ao,  we  have,  since  (/3  +  y)2  = 
—  2(1  + cos  a), 


—  AO-  = 


AO  = 


c  sm  b 


sni  A  +  sin  b  +  sni  c 

c  sin  B 
sin  A  +  sin  B  +  sinc 

c  sin  B 
sin  A  +  sin  b  +  sin  c 


2(1  + cos  A), 
V2(1  +  cosa), 
2  cos  ^  A. 


25.  If  tangents  he  draivn  at  the  veHices  of  a  triavgle  inscribed 
in  a  circle,  their  intersections  ivith  the  opposite  sides  of  the  triangle 
will  lie  in  a  straight  line. 


96 


QUATERNIONS. 


Let  o  be  the  center  of  the  cu-cle  (Fig,  54)  whose  radius  is  r, 

and  OA  =  a,   OB  =  (i,  oc  =  y.     Since  oa  and  ap  are   at  right 

angles, 

S(oa  .  ap)  =  0. 

But 

ap  =  AB  +  BP  =  AB  +  V/BC  =z  f3  —  a-\-l/{y  —  (3)  ; 


Fiff.  54. 


hence,  substituting  this  vahio  above, 
Sa[/3-a  +  .'/(y-^)]=0, 


and 


z/  =  - 


Say  —  iia/3 


Therefore 


OP  =  OB  +  DP  =  ^  +  .'/BC  =  f3-  f  +  ^f^^  (y  -  P) 

huy  —  »ap 

_(r+Say)/3-(/-"  +  Sa/3)y^ 
Say  —  Sa/3 

Similarly,  or,  by  a  C3X-lic  change  of  vectors, 

on-(>-'  +  Sa^)Y-(>"+S;8y)a 
Sa/?  -  S/?y  ' 

^,,^(r  +  S/3y)a-(r  +  Say)/? 
S^y  —  Say 
"WTience 

(Say  -  Sa/3)0P  +  (Sa/3  -  S/?y)oQ  +  (S/3y  -  Say)oii  =  0. 

But  also 

(Say  -  S:i(3)  +  (Sa^  -  S/3y)  +  (S/3y  -  Say)  =  0. 

Hence  p,  q  and  k  are  collinear. 

26.    The  sum  of  the  angles  of  a  triangle  is  two  rigid  angles. 


GEOMETEIC   MULTIPLICATION   AXD   DIVISI0:N-. 


97 


Let  a,  yS,  y  be  unit  vectors  along  bc,  ca  and  ab  (Fig.  55). 
Then  (Ai-t.  42) 


20 
a  iS 

-  =  e  " , 

y 

^      ^^ 


But 


/8" 


e", 


e    I     ^ 
7     ^     - 


2<;)      29      2i// 

1  =  e~^  e^  e  ~  = 


€"£"£"=  6" 


;(0  +  9  +  r^) 


Hence  -(0  +  ^  +  i/^)  =  an  even  multiple  of  2  (Art.  42) ,  as  2  7i, 

as  we  go  round  the  triangle  n  times. 

In  taking  the  arithmetical  sum,  or  passing  once  round,  we 
take  the  first  even  multiple  of  2,  or 


-  (c/,  +  ^  4- ^)  =  4  ; 

.-.     cjb  +  e  +  i/.=27r, 

and  the  sum  of  the  interior  angles  is  Stt—  27r  =  -,  or  two  right 
angles. 

27.    The  angles  at  the  base  of  an  isosceles  triangle  are  equal  to 
each  other. 

Let  a  and  /3  (Fig.  5G)  be  the  vector  sides  f^=-  ^• 

of  the  triangle,  and  Ta  ^  T,3.     Then,  if  the 
proposition  be  true, 


=  K 


or 


a-fS  13 -a 

a(a-  ^)-'=Kf3{(3  -  a)-'={l3  -  a)-% 
a(/3-a)  =  {a-f3)f3; 

2  C2 


which  is  true,  since  Ta  =  T/j. 


98 


QUATERNIONS. 


28,  To  find  a  point  on  the  base  of  a  triavrjle  s^ich  tJiat,  if  lines 
he  draivn  through  it  parallel  to  and  limited  by  the  sides,  they  will 
be  equal. 

Draw  DE  (Fig.  57)  and  df  parallel  to 
the   sides.     From    similar   triaugles,  if 

AE  =  XAC, 

AE         FB         AB  —  AF 


whence 


Now 


AC         AB  AB 

AF 

1  —  .T  = 

AB 

AD  =  AF  +  FD, 


or,  smee  fd  =  ae, 

=  ( 1  —  X)  AB  +  X\C. 

But,  since  fd  is  to  be  equal  to  ed, 

(1  —  a-)  Tab  =  .tTac  =  y  ; 

.-.     (1  — a:)TABUAB  =  ?/UAB, 

xTacUac  =  2/Uac, 

and  therefore 

AD  =  2/(Uab  +  Uac), 

and  D  is  on  the  angle-bisector, 

29.  If  any  line  be  draivn  through  the  middle  point  of  a  line 
joining  two  jparallels,  it  is  bisected  at  that 
point. 

30.  If  the  diagonal  of  a  parallelogram 
is  an  angle-bisector,  the  parallelogram  is  a 
rhombus. 

31,  In  an}'  triangle  the  sum  of  the 
squares  of  the  lines  gii,  ke,  df  (Fig,  58) 
is  three  times  the  sum  of  the  squares  of  the 
sides  of  the  triangle. 

32.    Tlie  sum  of  the  angles  about  tico  right  liiies  tchich  intersect 
is  four  right  angles. 


Fig.  58. 


GEOMETRIC   MULTIPLICATIOK   AND   DIVISION.  99 

33.  If  the  sides  of  any  polygon  be  produced  so  as  to  form 
one  angle  at  each  vertex^  the  sum  of  the  angles  is  four  right 
angles. 

34.  Find  the  eight  roots  of  tinity  (Art.  39).  ^ 

35.  The  square  of  the  medial  to  any  side  of  a  triangle  is  one- 
half  the  sum  of  the  squares  of  the  sides  tvhich  contain  it,  minus 
one-fourth  the  square  of  the  third  side. 

55.   Product  of  two  or  raore  Vectors. 
1.    Let  q  =  a/3,  r  =  y.     Then,  since  Sqr  =  Srg, 

Sa/3y  =  SyayS. 

Let  q  =  ya,  r  =  /3.     Then 

Sqr  =  ^rq  =  Sya/3  =  SySya  ; 
.-.    SaySy  =  S/3ya  =  Sya/3 (108), 

or,  the  scalar  of  the  2'>^'odiict  of  three  vectors  is  the  same  if  the 
cyclical  order  is  not  changed. 

This  ma}'  also  he  shown  b}"  means  of  the  associative  law  of 
vector  multiplication  as  follows  : 

ay5y  =  (a^)y  =  (Sa/5  +Ta/3)y. 

Taking  the  scalars 

Sa/3y=S(Sa^+Ta/3)y 

=  S(Va^  .  y),    since    S(Sa/3  .  y)  =  0, 
=  S  .  yTa^  ; 

introducing  the  term  S  .  ySa^  =  0, 

=  S  .  yYa/3  +  S  .  ySa;3 
=  S.y(Sa/3+Va^) 
=  Sy(a^)  =  Sya;8. 


100  QUATERNIONS. 

In  a  similar  manner 

Sa/3y=S.a(S/3y+Ty3y) 
=  S  .  a\Py 
=  S(y/3y.a) 
=  S(V/3y  +  S/3y)a 
=  S^ya, 


and,  as  before, 
2.  Again 


Sa,(3y=S/3ya=Sya/?. 

S«/3y  =  S.a(S/3y+V/3y) 
=  S  .  aypy 
=  -  S  .  aVy/3 
=  _Sa(Vy^  +  Sy/?); 
.♦.    Sa^y^-Say^ (109), 

or,  a  change  in  the  cyclical  order  of  three  vectors  changes  the  sign 
of  the  scalar  of  their  product. 

3.  Kesuming 

a,5y  =  a(/3y) 

and  taking  the  vectors, 

Ta^y  =  T.  a(S,^y+T^y) 
=  aS/3y  +V  .  aV/?y. 

VyiSa  =  V(Sy^+Vy/3)a 

=  Y.  aSy^-V.aTy^ 
=  V.  aSy;3+V.  aV^y 
=  y.a(Sy/?  +  y/3y) 
=  aSySy  +  Y  .  aVySy  ; 
.-.    Ta/3y=Vy/3a (110), 

or,  the  vector  of  the  product  of  three  vectors  is  the  same  as  the 
vector  of  their  jyroduct  in  inverted  order. 

4.  Geometrical  interpretation  of  Sa/Sy. 

Let  a,  /3,  y  be  unit  vectors  along  the  three  adjacent  edges  oa, 
OB,  oc  (Fig.  59)  of  an}-  parallelopiped,  6  being  the  angle  be- 


Also 


GEOMETRIC   MULTIPLICATION   AND   DIVISION.        101 

tween  a  and  /3,  find  6'  the  angle  made  b}'  y  with  the  plane  aob. 
Then 

a/3  —  —  cohO  +  e  sin6', 

e  being  a  vector  perpendicular  to  the  plane  aob. 
Oi:)erating  with  x  S  .  y 


Sa/?y  =  S (  —  cos 6  +  e  sin  ^) y 
=  S(sin^  .  ey). 

But   Sey  =  —  cos   of  the   angle 
between  e  and  y  =  —  sin^' ; 


Fi"-.  59. 


Sa/Sy 


sinO  sinO'. 


1/ j-'-^-' 


Now,  if  a,  ;8,  y  represent  as  vectors  the  edges  oa,  ob,  oc, 
whose  lengths  are  a,  h^  c, 

Sa/3y  =  -  TaT/iTy  sin  Q  sin  Q' 
=  —  «6csin^sin^' 


But  ab  sin  6  =  area  of  tlie  parallelogram  whose  sides  are  a  and 
h,  and  csin^'  =  perpendicular  from  c  on  the  plane  aob.     Hence 

—  Sa/3y  =  volume  of  a  parallelopiped  tvJiose  edges  are 
a,  b  and  c,  drawn  parallel  to  a,  /3  and  y. 

Cor.  1.  Whatever  the  order  of  the  vectors,  the  volume  is  the 
same  ;  hence,  as  already  shown, 

±  Sa/?y  =  ±  S/Sya  =  ±  Sya^  =  ip  Say/3,  etc. 

Cor  2.  If  Sa;8y  =  0,  neither  a,  ^,  nor  y  being  zero,  then  either 
^=0,  or  0'  =  0,  and  tlie  vectors  are  complanar. 

Cor.  3.    Conversely,  if  a,  ^,  y  are  complanar,  Sa^y  =  0. 

Cor.  4.  The  volume  of  the  triangular  pyramid  of  which  the 
edges  are  oc,  ob,  oa,  is  —  ^  Sa/3y. 

5.  We  have  seen  that  when  a,  (3  and  y  are  complanar,  Sa/8y=0, 
and  therefore  a/5y  is  a  vector.     To  find  this  vector,  suppose  a 


102  QUATERNIONS. 

triangle  constinictod  whose  sides  ab,  bc,  ca  have  the  directions 
of  a,  /3  and  y  respective!}',  a  vector  not  being  changed  b}-  motion 
parallel  to  itself.  Since  the  tensor  of  the  vector  sought  is  the  prod- 
uct of  the  tensors  of  a,  /8  and  y,  we  have  to  find  U(ab  .  bc  .  ca), 
i.e.,  its  direction.  Circumscribe  on  the  triangle  abc  a  circle  and- 
draw  a  tangent  at  a,  represented  b^-  t'at.  Since  the  angles  tab 
and  BCA  are  equal,  we  have 


CA  AT 


atJ 


whence 

TJ(bc  .  ca)  =  U(ab  .  at')  [=  U(ba  .  at)]. 

Introducing  Uab  x 

TJ(aB  .  BC  .  ca)  =  U(aB  .  AB  .  at')  [  =  U(aB  .  BA  .  AT)], 

or,  since      U(ab  .  ba)  =  — (U  .  ab)-=1, 

U(aB  .  BC  .  ca)  =  —  U  .  at'  =  U  .  AT. 

Hence,  if  a,  b,  c  are  ax^y  three  non-colUnear  points  in  a  plane, 
or  if  a,  /?,  y  are  the  sides  of  a  triangle  joining  them,  in  order 
(in  either  direction,  since  Va^y  =  Vy/3a) , 

a/3y,      /3ya,      ya/3 

are  the  vector  tangents  to  the  circumscribing  circle  at  the  angles 
of  the  triangle. 

Again,  if  a,  b,  c  are  any  three  points  in  a  plane,  not  in  a 
straight  line,  and  a  and  ft  are  two  vectors  along  the  two  succes- 
sive sides  AB,  bc  of  the  triangle  which  the}-  determine,  and  cd  a 
vector  drawn  from  c  parallel  to  y,  intersecting  the  circumscribed 
circle  at  d,  then  is  da  parallel  to  Va^y  =  8.     For 

8  =  a^y  =  a^^y  =  aft'^ft'^y  =  -  {TftYaft-^  =  -  (T^)^^y, 

whence  U  .  ^^^,  which  turns  /?  parallel  to  —  a,  turns  y  into  a 

direction  8  =  da,  the  opposite  angles  of  an  inscribed  quadi'ilateral 
being  supplementary. 


GEOMETEIC   MULTIPLICATION   AND   DIVISION.        103 

If  y  have  a  direction  siieli  that  CD  crosses  ab,  or  the  quacM- 

lateral  is  a  crossed  one,  it  is  evident  on  construction  of  the 

figure  that  „,  ,     , 

"^  U8'=  VajSy  =  U(ad)  =  -  U8. 

Hence  tlie  continued  product  of  the  three  successive  vector 

sides  of  a  quadrilateral  inscribed  in  a  circle  is  parallel  to  the 

fourth  side,  its  direction  being  towards  or  from  the  initial  point 

as  the  quadrilateral  is  uncrossed  or  crossed  ;  and,  converse^,  no 

plane  quadrilateral  can  satisf}'  the  above  formula  ±  US  =  Utt/?y, 

unless  A,  B,  c  and  d  are  con-circular.     The  continued  product 

of  the  four  successive  sides  of  an  inscribed  quadrilateral  is  a 

scalar,  for  ^..0,^0 

a/3yS  =  (a/3y)  8  =  ±S'  =  Td\ 

Since  the  product  of  two  vectors  is  a  quaternion  whose  axis  is 
perpendicular  to  their  plane,  while  the  product  of  a  quaternion 
b}^  a  vector  perpendicular  to  its  axis  is  another  vector  perpen- 
dicular to  its  axis,  and  so  on,  it  follows  that  the  continued 
product  of  any  even  number  of  complanar  vectors  is  generally  a 
quaternion  whose  axis  is  perpendicular  to  their  plane,  while  the 
product  of  any  odd  number  of  complanar  vectors  is  a  vector  in 
the  same  plane.     Hence  the  formulae 

Sa=0,      Sa/3y=0,      Sa/SySo"  =  0,      etc., 

for  complanar  vectors. 

If,  however,  the  given  vectors  are  parallel  to  the  sides  of  a 
pol^'gon  ABC MN  inscribed  in  a  circle,  then 


TJ(ab  .  bc  .  CD MN  .  na)=  U(ab  .  BC  .  ca)  TJ(ac  .  CD  .  da)  

X  U(aM  .  MN  .  Na). 

But  each  of  the  products  U(ab  .  bc  .  ca)  is  equal  to  U  .  at, 
at  being  the  tangent  to  the  circle  at  a.     Hence 

TJ(ab  .  bc  .  CD MN  .  na)  =  (TJ  .  at)", 

which  reduces,  according  as  n  is  even  or  odd,  to  ±  1  or  ±  U  .  at. 
Hence  the  product  of  the  vectors  wiU  be  a  scalar  or  a  vector 


104  QUATERNIONS. 

according  as  their  number  is  even  or  odd,  and  in  the  latter  case 
this  vector  is  parallel  to  the  tangent  at  a. 

If  the  vectors  are  not  complanar,  but  parallel  to  the  successive 
sides  of  a  gauche  polygon  inscribed  in  a  sphere,  the  polygon 
ma}'  be  divided  as  above  into  triangles,  for  each  of  which  the 
product  of  the  three  successive  sides  is  a  vector  tangent  to  the 
circumscribing  circle,  all  these  vectors  Ij'ing  in  the  tangent  plane 
to  the  sphere  at  the  initial  point.  If  the  number  of  sides  is  even, 
their  product  will  be  a  quaternion  whose  axis  is  perpendicular  to 
the  tangent  plane,  i.e.,  lies  in  the  direction  of  the  radius  of  the 
sphere  to  the  initial  point ;  if  odd,  the  product  is  a  vector  in  the 
tangent  plane. 

Hence,  if  a,  b,  c  and  d  are  four  given  points,  not  in  a  plane, 
AB  =  a,  BC  =  /S,  CD  =  y  being  given  vectors,  and  p  any  other 
point  such  that  dp  =  o-,  pa  =  p,  if  p  lies  on  the  surface  of  a 
sphere  through  the  four  given  points,  we  have  the  necessary  and 
sufficient  condition 

a/3y<Tp  =  poy^a, 

for  each  member  is  equal  to  minus  the  conjugate  of  the  other, 
and  must  therefore  (Art.  4G)  be  a  vector. 

6.    From  Equation  (56), 

fiy-y(3=2T^y. 
Operating  with  V .  a  x 

2y.aT^y=T.a(^y-y^). 

Introducing  in  the  second  member  fSay  —  /Say, 

=  T(a;3y-ay/S+/8ay-/3ay) 
=  \{a(3  +  /Sa)y  -T(ay/3  +  ya^) 
=  \.2{Sa(3)y-Yiay-\-ya)l3 
=  2ySa/3-2ySSay. 

Hence 

T  .  aV/3y  =  ySa/3  - /5Say       ....      (111). 


GEOMETRIC   MULTIPLICATION   AND   DIVISION.        105 

This  formula  may  be  extended.     Thus,  for  a  -write  TaS,  and 

we  have 

V  .  VaSV/3y  =  yS(Ya8)/3  -  /5S(Ya8)7, 

T  .  VaSVySy  =  ySaS/?  -  |8Sa8y  .      .      .      .      (112). 

An  inspection  of  this  formula  shows  that  it  gives  a  vector 
complanar  with  y  aud  /S.     Moreover,  since 

T  .  TaSTySy  =  V  .  Vy)8Va8  =  SSy^Sa  -  aSy/3S, 

it  is  also  complanar  with  a  and  8,  and  is,  therefore,  parallel  to 
the  line  of  intersection  of  the  planes  of  a,  8,  and  /?,  y. 
Similarl}' 

y.V/5yVaS=SS/3ya-aS|8y8  =  -V.Ta8Y^y  .      (113). 
Adding  Equations  (112)  and  (113) 

8S^ya  -  aSy8y8  +  ySa8/3  - /3SaSy  =  0     .      .      (114), 

or 

8Sa/Sy  =  aS^y8-^Say8+ySa^8       .       .       .      (115). 

a  formula  expressing  a  vector  8  in  terms  of  any  three  given  di- 
plauar  vectors,  a,  f3,  y;  so  that,  if 

S/3y8  —  b,       —  Say8  =  SyaS  =  C,      Sa^8  =  a,      Sa^ffy  =  m, 
8  =  7}l~^  (ba  +  c^  +  ay) . 

7.   Resuming  Equation  (HI),  and  adding  aSySy  to  both  mem- 

V  .  aT/3y  +  aSySy  =  ySa^  -  /8Say  +  aS/3y , 
whence 

T.a(S/3y  +  V/3y)  = 

Ta/?y  =  o.S/3y  -  ySSay  +  ySa^  .      .      .      .      (110). 

The  form  of  this  equation  shows  that  a  and  y  may  be  inter- 
changed, or  that  \a./3y  =  Vy/?a,  as  ah'eady  shown. 
Again,  replacing  a  by  \a^  in  Equation  (111), 

V  .  Ya/iYfSy  =  yS(Va/S)^  -  /8S(Va^)y, 

or 

T  .  Ytt^V/5y  =  - /5Sa^y (117). 


106  QUATERNIONS. 

8.   TVriting  VySa^  fii'st  as  T(y  .  Sa(3),  and  then  as  y(yS  .  a^), 
we  have 

V(y  .  Baft)  =  T .  y(S8a(3  +Y8a(3) 

=  ySSayS  +Y  .  yT8a(3 

[Equation  ( 1 1 G )  ]      =  ySa/3S  +  TySSayS  -  VyaS8/3  +  Yy;8S8a.     (a) 

y(yS  .  a/?)  =T(Sy8  +Ty8)  (Sa;8  +Va/3) 

=  Ty8Sa/3  +Vay8SyS  +V  .  YySYaft 

=  Yy8Sa;8  +Ya^Sy8  - Y  .  Ya;8Yy8, 
or,  Equation  (112), 

=  Yy8Sa;8  +  Ya/3Sy8  -  8Sa/?y  +  ySaySS.        (b) 

Equating  (a)  and  (&), 

8Sa/Sy  =  Y/3ySa8 +YyaS/?8 +YaySSy8     .      .      (118), 

a  formula  expressing  a  vector  8  in  terms  of  three  other  vectors 
resulting  from  their  products  taken  two  and  two  ;  so  that,  if 
Sa^y  =  m,  Sa8  =  a,  S^8  =  b,  Sy8  =  c, 

8  =  m-^  (aY/5y  +  &Yya  +  cYayS) . 

Operating  on  Equation  (118)  with  S  .  p  x ,  we  obtain,  since 
S  •  p^  ya  =  S/aya, 

SpSSaySy  —  S/38S/)ya  —  Sa8Sp;8y  —  Sy8Spa/3  =  0, 
or 

•       Sa8SpySy  -  S/?8Sypa  +  SySSpayS  -  Sp8Su/?y  =  0  .      (119), 

a  formula  eliminating  8. 

56.   Exercises. 

Prove  the  following  relations : 

1.  Sa/3y8  =  SSaySy. 

2.  aft.  I3y  =  - ay. 

3.  a-ft-=aft  .  /3a. 

4.  S  .  Ya)8Y^y  =  S  .  a^Y/3y (120). 

5.  Sa/Sy8  =  Sa^Sy8  -  SayS^8  +  SaSSySy (121), 

from  which  show  that  Safty8  =  S/3y8a. 


GEOMETRIC   MULTIPLICATION   AND   DIVISION.        107 

6.  S  .  Ya/3Vy8  =  Sa8S/?y  -  SayS/?S (122). 

7.  S(a  +  ;8)(^  +  y)(y  +  a)  =  2Sa^y. 

8.  a^y  +  yy8a  =  2Vay8y. 

9.  apY-y(3a=2Saj3y. 

10.  V(aY^y  +  /5Vya  +  yVa/5)=0 (123). 

11.  Va^y  +  Vya^=2ySa/3. 

12.  aV/3y  + /SVya  4- yVa/3  =  3  Sa/Sy. 

13.  S  .  Va/3V|8yYya  =  -  (Sa^y)2. 

14.  S  .  yV^a  =  y;8a  -  yS/?a  +  ^Sya  -  aS/3y        .      .      .      (124). 

15.  S  .  Y{Ya(3Y(3y)  V(V/5yVya)  V(VyaVa^)  =  -  (SafSyY. 

16.  S  [VaySVyS  +  VayVS/?  +  VaSV^y]  =  0       ....      (125). 

17.  If  Sa^y  =  m,  Sap  =  0,  S/3p  =  0,  Syp  =  0,  sliow  that  p  =  0. 

Conversel}",  if  p  is  not  zero,  then  Sa/?y  =  0. 

18.  Interpret  p  =  a"'/?a. 

We  have  first,  directly, 

Sap|8  =  Saa-i/3a/3  =  S/3a^  =  S/S^a  =  0  ; 
.'.  p,  a  and  /5  are  complanar. 

Sap  =  Saa~^y8a  ^  S/3a, 
-  TpTa  COS  ^  =  -  TaT/3  COS  </), 

or,  since  Tp  =  T/S,  cos^  =  cos  <^. 

Similarly  Vap  =  V/5a,  and  sin^  =  sin  ^.     Hence 


and  a  bisects  the  angle  between  (3  and  p. 

19.  Show  that  p  =  afSa-^  =  a'^Sa^  -  Va/5) . 

20.  p  being  any  vector,  show  that  V  .  YapYpfS  =  xp. 

21.  If  SayS  =  — a^,  show  that  a  is  perpendicular  to  /5  —  a. 

(3         13^ 

22.  "What  are  the  relative  directions  of  a  and  B,  if  K-  = ? 

)8       )8  '^  a  a 

If  K-=-? 


108 


QUATERNIONS. 


57.   Examples. 

1.    The  aUitudes  of  a  trianrjh  intersect  in  a  jyoint. 

Let  (Fig.  GO)  ac  =  /S,   en  =  a,   ar  =  y. 
Fig.  GO.  Then  vectors  along  c'c,  n'n  and  a'a  are 

c 

iA' 

M        rospeetivel}'.     Now 


B    or 


AG  =  AC  +  CO  =  AB  +  liO, 
I3-X€y  =  y+  y€(3. 


Operating  with  x  S  .  /?,  we  have,  since  ySeft-  =  0, 

^  ^        Sa/3 
Heyft 

Having  assnmed  o  to  be  the  intersection  of  the  altitudes  bb' 
and  cc,'  let  o'  be  the  intersection  of  a  a'  and  cc.     Tlien 

AO'  =  AC  +  CO^ 

or 

Zea  =  13  —  x'ey. 

Operating  with  x  S  .  a 


Seya        Suey 

^  S/?a  ^      Sojg 

S(y-^)ey       -S(3ey 

^  Sr,/? 

Sey/3' 


Hence  o  and  o'  coincide,  and 


^0  =  ^4-— ^€y. 
Sey/^ 


GEOMETEIC   MULTIPLICATIOiSr   AND   DIVISION.        109 

2.    To  circumscribe  a  circle  about  a  triangle. 
Let  (Fig.  61)  AC  =  (3,  en  =  a,  au  =  y. 
Then 

a'o  =  —  Xea, 
C'o  =  yey, 
b'o  =  —  Z€(3. 

Operating  with  x  S  .  /S  on  tlie  expres- 
sion 

AO  =  ^y  +  yey  =  -h  13  —  Zeft, 

we  have 

y  = tl_. 

2  Hey (3 

Operating  with  x  S  .  a  on 

bo'  =  —  ^y  -\-  y'ey  =  —  ha  —  xUa, 

we  have 

^        2Seya  2  Ssy/? 

Therefore  y  =  y'  and  o  and  o'  coincide. 
The  radius  ma}'  be  found  bj'  squaring 

AO  =  iy+,.y  =  iy-^ey, 
whence 

O  0      07  9  0  _ 

2_      c-      c-fro-cos-c 
4       4  6Vsin^A 

since,  if  a,  b,  c  are  the  tensors  of  a,  /5,  y, 

o.     so  9  o-/r COS- c 

4  0-c-  sur  a 
Hence 

Vc-  sin-  A  +  a^  cos"  c  a 


R  = 


2  sin  A  2  sin  a 


110 


QUATERNIONS. 


3.  In'any  trianrjle^  the  centre  of  the  circumscribed  circle,  the 
intersection  of  the  altitudes  and  the  intersection  of  the  medials  lie 
in  the  same  straight  line;  and  the  distance  betioeen  the  last  tv:o 
points  is  tioo-thirds  of  the  distance  betioeen  the  first  tico. 

Let  M  (Fig.  62)  be  the  intersection  of 
Fig.  C2.  the  medials,  a'  that  of  the  altitudes,  and 

c  the  center  of  the  circle. 

Then,  from  Ex.  5,  Art.  11,  where  cp 
(Fig.  11)  is  given  in  terms  of  the  adjacent 
sides,  we  have 


AM  =  ^(/3  +  y). 
From  Ex.  1,  Art.  57, 


From  Ex.  2,  Art.  57, 

But 

and 


I        o    ,    Sap 

w  Sa/3      . 


CM 


AM  -  AC  =  -J/^  —  •  y  +  i^-^fy, 


Sa/3  ^.. 


Sa/3 


MA'  =  AA'-AM=|^-iy  +  i^ey- 
.-.  ma' =  2 CM, 

and,  since,  as  vectors,  they  are  multiples  of  each  other,  and  have 
a  common  point,  they  form  one  and  the  same  straight  line. 

4.    To  find  the  condition  that  the  perpendicidars  from  the  angles 
of  a  tetraedron  to  the  opposite  faces  shall  intersect. 

AVith  the  notation  of  Fig.  52,  the  perpendiculars  from  a  and  b 
on  the  opposite  faces  are 

V^y     and    Tya. 

If  they  intersect,  at  p  say,  then  must  a,  b,  p  lie  in  one  plane. 
Hence,  Art.  55,  4,  Cor.  3, 

S[(^-a)Vy8yVya]=0, 


GEOMETEIC    MULTIPLICATIOISr   AND   DIVISION.        Ill 

or 

S  (^  -  a)  [S  .  V^yVya  +  V  .  V/^yVya]  =  0, 

S(/5-a)V  .  V/3yVya  =  0. 

Fig.  52  (Ms). 

But,  Equation  (117),  ^ 

V  .  V^yVya  =  -  yS/?ya  ;  //    \ 

.•.-(S;8y-Say)S/Sya=0,  /     /         \ 

or  /  /  \ 

S/3y=Say.         (ci)        o^- / -^B 

From  the  figure,  we  have  ^^L-"^"^ 

A 

BC-  +  OA"  =  (y  —  jSy  +  a^ 

or,  from  (a) ,  =  y-  —  2  Say  +  (3-  +  a^ 

=  (y-a)^  +  ^^ 
=  AC^  +  OB^. 

Hence  the  condition  is  that  the  simis  of  the  squares  of  each  pair 
of  opposite  edges  shall  he  the  same. 

5.   Interpret  Equation  (118), 

8Sa/5y  =  V/3ySa8  +  VyaS/S8  +  Va/3SyS, 

under  the  condition  that  a,  /S,  y  be  complanar  with  8. 

If  a,  ^,  y  are  complanar,  Sa^y  =  0,  and  therefore,  8  being  in 
or  out  of  the  plane, 

Sa8V/?y  +  S/3SVya  +  Sy8Va/?  =  0.  (a) 

If  8  be  in  the  plane,  we  have  for  any  four  co-initial  lines 

OA,  OB,  OC,  CD, 

sin  BOG  COSAOD  +  sin  CO  A  cos  bod  +  siuAOB  cos  COD  =  0, 

and,  for  a  line  perpendicular  to  od, 

sinBOC  sinAOD  +  sincoA  sin  bod  +  sinAOB  sin  cod  =  0. 

If  8  is  perpendicular  to  the  plane,  the  terms  in  (a)  vanish 
separately. 


112  QUATERNIONS. 

6.  If  X,  Y,  Z  he  the  angles  made  by  any  line  op  idtli  three 
rectangular  axes,  then 

cos-  X  +  cos^  Y  +  cos- Z  =\. 

From  Equation  (07) 

ip  =  xi-  +  yij  +  2ik  =  —  x  +  yl'  —  zj, 
whence 

Operating  in  a  similar  manner  with  S  ../  X  and  S  .  ^'  X  we  obtain 

-p'  =  {Hipy-  +  {Sjpr-  +  {skpy. 

If  Tp  =  r,  then  p-  =  —  r^,  Sip  =  —  r  cos  X,  etc.     Hence 

op^  =  op-  (cos-  X  4-  cos-  Y  +  cos-  Z) , 

or 

cos^  X  +  cos-  Y  +  cos^  Z  =  1 . 

Applications  to  Spherical  Trigonometry. 

Let  ABC  (Fig.  G3)  be  an}-  spherical  triangle  on  the  surface  of 
a  unit  sphere  whose  center  is  o  ;  a,  /?,  y 
being  unit  vectors  from  o  to  the  vertices. 
The  sides  ab,  bc,  ca  represent  vei'sors 
c  whose  angles  are  c,  a,  b,  and  axes  are 

^'f\\      p/     OC'=y;     OA'  =  a;     Ob'=^';      a',     f3',     y' 

^h\n        l)eing  unit  vectors  to  the  vertices  of  the 
P^n       polar  triangle  whose  sides  are  a',  b',  c', 
the  supplements  of  the  opposite  angles 
A,  B,  c  of  the  triangle  abc. 

7.  We  have  first  o      o 
^  =  ^-.  (a) 
7       "7 


GEOMETRIC   MULTIPLICATION  AND   DIVISION.       113 
Taking  the  scalars,  we  have  [Equation  (90)], 


But 


and 


^         13     a  B     a 

s-  =  s- S-  +  S  .y-Y-. 

y  "7  ^7 

13  f3  a 

S-  =  cosa,     S-  =  cosc,      S-  =  cos&, 
7  «  7 

13    a 
S  .  T'-Y-  =  S(sinc  .  y')(sin6  .  /?') 

=  sine  sin&  Sy'yS' 

=  —  sin  c  sin  b  cos  a' 

=  sine  sin  &  cos  a. 
Hence,  in  («), 

cos  a  =  cos  c  cos  b  +  sin  c  sin  b  cos  a. 


By  a  cychc  permutation  of  tlie  letters  in  (a),  we  obtain 

a      p    o. 
Whence,  as  before 

sZ  =  s^s^  +  s.y|y^, 

a  p     a  pa 

or 

cos  b  =  cos  a  cose  +  sin  a  sine  Sa'y', 

in  which  Sa'y'  =  — cos6'  =  cosb. 

.  • .  cos  b  =  cos  a  cos  e  +  sin  o  sin  e  cos  b  .  (c) 

Similar!}',  or  directly  b}-  c3-clic  permutation  in  (e), 

cos  e  =  cos  b  cos  a  +  sin  b  sin  a  cos  c. 

From  the  relation 

/3'_/3'   a' 
y'-a'y' 

may  be  deduced  in  like  manner 

—  cos  A  =  cos  c  cos  B  —  siu  c  sin  b  cos  a. 


114  QUATEEXIONS. 

8.   Resuming  the  equation 

y    «  y 

of  the  last  example,  and  taking  the  vectors,  we  have  [Equa- 
tion (91)], 


But 


y  a     y  y      a.  "7 

V-  =  —  a  sma, 

y 

/8     a, 

S- V-  =  cose  (/3  sin 6)=  cose  sin 6  .  fi, 

S-  T  -  =  cos  h  (y'  sin  e)  =  cos  &  sin  c  .  yl 

y      a  \/  /  17 

13     a 
T  .  V - V-  =  T(y  sine)  (fi'sinb)  =  sin e  sin 6  Yy'yS' 

y 

=  sine  sin?>(— asina')  =  —  sin  e  sin  &  sin  a  .  a. 
Substituting  in  (a) , 

—  siua  .  a'=cose  sin&  .  /8'-fcos6sinc.y'— sincsin6sinA.a.  (b) 
Operating  with  x  S  .  y'~S 

—  sina  .  S-,  =  cose siu6 S^ +cos6 sine S—,— sine sin&slnAS-:) 

y  y  y  y 

in  which 

S  — ,  =  cos  b'  =  —  cos  B, 

y 

S  — =  — COSA, 

y 

S  — ,  =  0,  since  a  and  y'  are  at  right  angles. 

Hence 

sin  a  cos  B  =  cos&  sine  —  cose  sin  6  cos  a, 


GEOMETEIC   SnjLTIPLICATIOlSr  AilSTD  DIVISION.       115 
and  in  the  same  manner,  or  hj  a  cyclic  permutation  of  the  letters, 

sin  6  cos  c  =  cos  c  sin  a  —  cos  a  sin  c  cos  b  , 
sin  c  cos  A  =  cos  a  sin  b  —  cos  b  sin  a  cos  c. 

9.    Operating   on    Equation   (6)   of   the    last   example   with 
X  y .  y'-^  instead  of  X  S  .  y'-\ 


— sinaV 


But 


V- 
7 


B'  ' 

=  coscsin&V  — +  COS&  sine  V— ,  —  sine  sin&  siuAV— ,• 


=  jSsinb'  =  ySsiuB, 

=  —  a  sin  a' =  —  a  sin  A, 

=  0. 


Fig.  63. 


Substituting  these  values 

—  sin  a  sin  b  .  /3  =  —  cos  c  sin  b  sin  a  ,  a 
—  sin  c  sin  b  sin  a  .  V 

Operating  with  X  a~  ^ ,  and  substituting  for 


7' 


-  =  cos  c  +  y  sm  c, 


we  obtain 


or 


—  sin  a  sin  b  cos  c  —  sin  a  sin  b  sin  c  .  y '  =  —  cos  c  sin  b  sin  a 
—  sin  c  sin  b  sin  a  .  y'. 

Equating  the  scalar  or  vector  parts,  we  have  in  either  case 

sin  a  sinn  =  sin  a  sin&, 

sina  :  sin&  : :  siuA  :  siuB. 


The  formulae  of  the  preceding  examples  have  all  been  deduced 

from  the  equation  -  = The  product  as  well  as  the  quotient 

may  also  be  employed,  as  follows : 


116  QUATERNIONS. 

10.  Assuming  the  A^ector  product 

Ya;QV/?y, 
and  taking  the  vector  part,  we  have  [Equaliuii  (117)], 

V  .  \a(3\/3y  =  -  fSSa/Sy.  (o  ) 

But 

V  .  Ya/^Y/Sy  =  y(7'sino)  (a' sin  o.)=  sine  sinrt  sins  .  f3, 

and,  Art.  55,  4, 

Sa/8y  =  —  sine  sin^,' 

6'  being  the  angle  made  by  oc  with  the  plane  of  c.     Substituting 

ill  (c)^ 

sine  sino  sinii  .  (3=  sine  sin^' .  (3, 

or 

sin 6'  =  sino  sine. 

B3-  permutation,  from  («), 

^---U'  y  ,  \ya\ afS  =  -  aSya(3  =  -  aSa/?y, 

T^-^ii      or 
c  siu6  sine  siuA  .  a  =  sine  sin^' .  a, 

.'.  sin^'  =  sin  6  sin  a. 

Equating  these  values  of  sin^'  we  have,  as  in  Example  9, 
sin  a  :  sin  &  : :  sin  A  :  sin  B. 

11.  Let  p„,  Pj,  2^c  represent  the  arcs  drawn  from  the  vertices 
of  ABC  perpendicular  to  tlie  opposite  sides. 

Resuming  Equation  (a)  of  the  preceding  example,  and  taking 
the  tensors, 

TV  .  Ya(3\/3y  =  Sa/3y  =  sinc  sin/)„ 
=  Sf3ya  =  sin  a  sinp„, 
=  S/a/3  =  sin  b  sin^9j, 


GEOMETRIC    MULTIPLICATION   AND   DIVISION.        117 

and,  taking  the  tensor  of  V  .  Ya/SY^y  from  the  last  example, 

sine  sin  a  sinB  =  sin  a  s'mj^a  =  sin 6  sinj^j  =  sine  sinp^, 
or 

sin  Pa  =  sine  sinB, 

sin  c  sin  a   . 

sni  2h  = sin  b  , 

sin  6 

sinpc  =  sin  a  sinB. 

12.  SJioio  that  if  abc,  a'b'c'  be  two  tri-rectangular  triangles  on 
the  surface  of  a  sphere^ 

cosaa'  =  cosbb'coscc'  —  cosb'c  cosbc^ 

the  triangles  being  lettered  in  the  same  order. 

Let  a,  j3,  y,  a',  (3',  y'  be  the  vectors  to  the  vertices.  These 
being  at  right  angles,  in  each  triangle,  we  have 

cos  Aa'  =  -  Saa'  =  -  S  .  YfSyYfS'y', 

or.  Equation  (122), 

cos  aa'  =  S^^'Syy'  -  ^/3'yS/3y' 

=  COS  bb'  cos  cc'  —  COS  b'c  cos  BC' 

[The  vectors  of  Equation  (122)  are  arbitrary,  but  we  may 
divide  both  members  by  the  tensor  of  the  product  of  the  vectors, 
so  that 

S(VUay8VUyS)  =SUaSSU^y  -  SUaySU/3S, 

for  the  unit  sphere.] 

13.  Let  ABCD  be  a  spherical  quadrilateral  whose  sides  are 
AB  =  a,  BC  =  b,  CD  =  c,  DA  =  d,  the  vectors  to  the  poles  of  these 
arcs  being  a,  /3J  y',  8'  respectively-.     Then 

Va/8  =  a' sin  a, 
VyS  =  y'sinc. 


118  QUATERNIONS. 

From  Equation  (122), 

S  .  YafiJyB  =  SaSS^y  -  SayS/?S, 
or 

sin  a  sin  c  Sa'y'  =  ( —  cos  da)  ( —  cos  bc)  —  (  —  cos  db)  ( —  cos  ac)  . 

But  Sa'y'  =  — cosL,  L  being  the  angle  formed  by  the  arcs  ab 
and  CD  where  they  meet,  the  arcs  being  estimated  in  the 
directions  indicated  by  tlie  order  of  their  terminal  lettei's. 
Hence 

siuAB  sin  CD  cosl  =  cosac  cosbd  —  cos  ad  cosbc, 

a  formula  due  to  Gauss. 

14.  Retaining  the  above  notation,  abcd  being  still  a  spherical 
quadrilateral,  denote  the  angles  at  the  intersections  of  the  arcs 
ab  and  cd,  ac  and  db,  ad  and  bc,  by  L,  m  and  n  respectively. 
Then,  from  Equation  (125), 

S[Ta/3Vy8  +  VayYS/3  -|-  Ya8T/3y]  =  0, 

we  have  identically 

siuAB  sin  CD  cosl  +  sin  ac  sIubd  cosm  +  sin  ad  siuBC  cosn  =  0. 

Were  the  points  a,  b,  c,  d  on  the  same  great  circle,  the  angles 
L,  M  and  N  would  be  zero,  and  the  above  reduces  to 

sinAB  sin  CD  +  sin  ac  siuBD  +  sin  ad  siuBC  =  0, 

and  for  a  line  oaJ  perpendicular  to  oa  and  in  the  same  plane, 
dropping  the  accent,  we  have 

cosAB  sin  CD  +  cos  AC  siuBD  +  cos  AD  sinBC  =  0, 

which  are  the  results  of  Example  5  of  this  article. 


GEOMETRIC   MULTIPLICATION   AISTD   DIVISION.        119 

58.   General  Formulae. 

1.  We  have  seen,  Equation  (86),  that  SS  =  2S  and  T2  =  2Y ; 
but  (Art.  50,  4)  that  2T  is  not  equal  to  T2,  nor  2U  to  US.  We 
have  also  seen.  Equations  (9G)  and  (97),  that  Tn  =  nT  and 
Un  =  nU;  but  Sn  is  not  equal  to  ns,  nor  Vn  to  HV :  for, 
1st,  sn  is  independent  of  the  factors  under  the  11  sign,  provided 
the  product  remains  the  same,  while  ELS  is  dependent  upon 
them  ;  and,  2d,  because  (Art.  55,  5)  nV  is  not  necessarily  a 
vector. 

2.  Resuming  Equation  (92), 

Srg  =  Sgr, 

and,  since  r  is  arbitrary,  writing  rs  for  ?•,  we  have,  by  the  asso- 
ciative law  (Art.  52), 

S(rs)g  =  Sg(rs), 
Sr(sg)=S(sg)r, 
.-.  Srsg=Ssgr=Sgrs  .     .     .     .     (126), 

a  formula  which  may  evidently  be  extended.  Hence,  tJie  scalar 
of  the  product  of  any  number  of  quaternions  is  the  same,  so  long 
as  the  cyclical  order  is  maintained. 

3.  Let  J)-,  g?  '%  s  be  four  quaternions,  such  that 

qr=ps.  (a) 

Operating  with  Kg  x  , 

Kg  .  qr  =  (Kg  .  q)r  =  (gKg)?*  =  Kg  .  ps, 

since  conjugate  quaternions  are  commutative.     Hence 

(Tg)^r  =  Kg  .  ps, 
or 

Kg  .  ps  1 


Operating  on  (a)  with  xKr,  we  have 
qr  .  Kr  =  ps  ,  Kr, 


Rg.j?s  =  -»ps    •     •     •     (l^'^)' 


120  QUATERNIONS. 

or 

q{Try-  =  p«Kr, 

psKr  1 

.*.  9  =  ^'  =  i^«R'-=/>s-      .     .     .     (128). 

Hence,  in  any  eq^iation  of  the  jjroduds  of  two  quaternions, 
the  first  factor  of  one  member  may  he  removed  by  loriting  its  con- 
jxigate  as  the  first  factor  of  the  second  member,  and  dividing  the 
latter  by  the  square  of  the  tensor,  or  simply  by  introducing  the 
reciprocal  as  the  frst  factor  in  the  second  member.  )iy  substi- 
tuting the  word  last  for  first,  the  above  rule  will  appl}'  to  the 
second  transformation. 

4.  Resuming,  for  facilit}'  of  reference,  the  equations 

9  =  1  =  ^  (cos<^  +  £sin<^)  =  Tq  .  Vq  =  Sq  +  Yg,     (A) 

1      /S      T^ 
fy-i  =  -  =  -  =  ^(cos<^-esiu</)),  (B) 

rp 

Kry  =  —  (cos  4>  -  £  sin  4>)  =  Sq-  \q,  (C) 

we  observe  directly  that 

Sq  =  S{Tq.Vq)  =  Tq.SVq     .      .      .      (129), 

Tg  =  TVry  .  V\q  =  1q  .  Tllry      .      .      .      (130) , 

TVg  =  Try  .  TYU5  =  TVKg     ....     (131). 

5.  It  has  been  alread}'  shown  (Art.  54,  Fig.  40)  that 
(Ta)2-|-(Tj8)-=(Ty)-,  and  (Art.  54,  Fig.  42)  that  Ta=Ty.cos<^, 
T/3  =  Ty  .  sin  <^  ;  and  therefore 

(Ty)=^COS=^<^  +  (Ty)=^  sin-0  =  (Ty)S 
or 

sin-^  +  cos-</>  =  l. 

Hence,  from  Equations  (44), 

(SU(/)2  +  (TTLVy)-  =  l     ....     (132). 


GEOMETRIC    MULTIPLICATION   AND   DIVISION.        121 

This  important  formula  might  have  been  written  at  once  by 
assuming  the  above  well-known  relation  of  Plane  Trigonometry. 

6.   From  Equations   (129)   and   (131),  we  may  write  Equa- 
tion (132)  under  the  form 

(Sry)-  +  (T\V7)-  =  (Tg)2 (133), 

or,  from  Equation  (107), 

{iiqr-{\qr  =  {Tqy  =  (^qy-h{T\qy       .     (134), 


since  e^  =  —  1 . 

7.    Comparing  (A),  (B)  and  (C), 

SUg  =  SU-  =  SUKg    .     . 

.     (135), 

1 
T  VUg  =  T  VU  -  =  T  VUKg       . 

.     (136), 

and  from  Equations  (129)  and  (135), 

1 

S(/  =  Tg.  SUg  =  Tg.  SU-  =  Tg.  SUKg.     .     (137). 

8.    Since  Tg  =  TKg,  we  have 

Tg.TKg  =  (Tg)2 (138), 

and  Tg  being  a  positive  scalar, 

KTg  =  TKg (139). 

As  exercises  in  the  transformation  of  these  and  the  following 
S}Tnbolical  equations,  some  of  the  results  alread}^  obtained  will 
be  deduced  anew.  Thus,  to  prove  that  T(gg')  =  TgTg',  whence 
T.g2  =  (Tg)2,  we  have 

(Tgg')-=(gg')K(gg')  Equation  (107) 

=  gg'Kg'Kg  Equation   (99) 

=  g(g'Kg')Kg  =  (Tg')2gKg 
=  (Tg')2(Tg)2, 

.-.  Tgg'  =TgTg: 


122  QUATERNIONS. 

9.  Siihstitnting  for  Sq  and  T\q  their  values  from  Equations 
(79)  and  (131) 

{SKqy-  +  {T\Kqy  =  {iiqy-  +  {'r\qy-  .    .   (uo). 

10.  Resuming  from  Art.  51,  1,  the  expressions 

Yrq  =  Sr\q  +  SryYr  -\-  V  .  \rYq,  (a) 

\qr  =  HqYr  +  ^r\q  +  V  .  YqYr,  (6) 

Sgr  =  S(/Sr  +  S  .  VgVr,  (c) 

we  have,  hy  adding  and  subtracting, 


V(//-  —  \rq  =  2  V  .  \q\r 
And,  if  (/=  r,  from  (a)  and  (c), 

\qy  \ 


(142), 


V.  q-  =  2Sq\q 
S.f/  =  (Sg)=  +  (V. 
whence 

g2  =  (Sg)2  +  2S(/V5  +  (Yg)2.     .     .     (143). 


Dividing  Equations  (142)  by  (Tg)^ 

SU.r/  =  (SU5)^+(VUg)M  .144. 

TU  .  fy-  =  2  SUg  .  VUg         )     '     '     * 

since,  evidently, 

S.r/  =  (Tg)^SU.f/)  ,145. 

V.fy-  =  (Tr/)2VU.r/j 

Again,  substituting  in  the  second  of  Equations  (142)  the  value 
of  {\qy  from  Equation  (134),  we  have 

S.q'  =  2{Sqy-{Tqy (146), 

and  dividing  by  (Tfy)- 

SU.f/=2(SUry)2-l (147). 

Substituting  (Sg)-  from  the  same  equation 

S.^-  =  2(Vry)2  +  (Try)2 (148). 


GEOMETRIC   MULTIPLICATION   AND   DIVISION.        123 

Equations  (146)  and  (148)  may  be  written 

(S  +  T)g2=2(Sfi)2     and      (S  -  T)r/=  2(Vg)'. 

Introducing  in  (a),  or  (6),  tlie  condition  that  q  and  r  are 
complanar,  we  have,  after  substituting  versors, 

yUgr  =  VUfiSUr  +  VUrSUg, 

since,  under  the  condition,  V(VU5VUr)  =  0. 

Taking  the  tensors,  since  q  and  r  are  complanar, 

TVUgr=TVUgSUr  +  SUgTVUr  .     .     .     (149), 

and,  interpreting,  Art.  51,  6, 

sia(^  +  ^)  =  sin^  cos</)  +  cos^  sin</). 

Introducing  the  same  condition  of  complanarity  in  (c) 

Sg?"  =  SryS^-  -  TVf/TVr, 

or,  substituting  versors  as  above, 

SUgr  =  SUgSUr  -  TVUgTVUr    .     .     .     (150), 
or,  interpreting, 

cos  {0  -\-<^)=-  cos  9  cos  (^  —  sin  <^  sin  0. 

11.    Putting  Equation  (146)  under  the  form 

[S.g^  +  T.g^ 
^^  =  \ 2 ' 

and  writing  Vg  for  g,  we  have 


sVg  =  Vi(Sg  +  Tg)      ....     (151). 

12.  Taking  the  tensors  of  the  fii'st  of  Equations  (142) ,  we  have 

TV  .  (f 


124  QUATERNIONS, 

and  writing  V^  for  q 

TYq 

TV.Vy  = 


2SV^ 
or,  by  Equations  (133)  and  (151), 

UTqy-i^cjY 


TV 


whence 
and 


TV.  Vr7=Vi(T9-S(/) (152), 


13.   From  the  definition  of  the  powers  of  a  quaternion,  we  liavc 
q-q-^h     {q"''T  =  q"'     ....     (154). 
Hence,  since  q  =  Tq  ,  Vq,  TH  =  HT  and  UIT  =  nu, 

Tfy-""  .  Tfy'"=l,     TJ^?""*  .  U^"  =  1       .     .     (155). 
Also,  because  IVy""'  =  UK^"*, 

q-^=Tq-'^  .  U^?-"*  =  Tg-'"  .  UKry"  =  Tg--'"Kr/'», 
or,  since  Kj)^  =  KgKp,  writing  j)^  for  7,  and  making  ??i  =  1, 

(i^9)"'  =  T(i)g)-=^Kp7  =  1{pqY-KqlLp 
=  1{pq)-\1q)\Ti,yq-'p-\ 

Or 

(i?5)-^  =  y-'i>-^ (150), 

the  reciprocal  of  the  jyrodiict  of  tioo  quaternions  being  equal  to  the 
product  of  their  reciprocals  in  inverted  order. 

This  formula  may  be  extended  by  the  Associative  principle,  hy 
a  process  siniihu-  to  that  employed  in  the  deduction  of  Equation 
(126),  so  that  if  11'  represent  the  product  of  tlie  same  factors  as 
those  of  n,  in  reverse  order, 

{Uq)-'  =  U'q-'       .....      (157). 


GEOMETRIC   iniLTIPLICATION   AND   DIVISION.        125 

The  equation  'Kpq  =  'KqE.p  may  be  deduced  without  reference 
to  spherical  arcs.     For,  by  Art.  44,  any  two  quaternions  can  be 

reduced  to  the  forms  g  =  — ,  P  =  -,■>  whence 
a  p 

PQ.  =  ^^     or    2^Q  .  a  =  y,    P/3  =  y, 
and  therefore 

Kp  .  y  =  Kp  .  i?/?  =  (Kp  .  p)[i  =  (Tjjy-fS. 

Now 

(KgK2))y  =  Kg(Tp)2/3  =  (Ti;)-Kg  .  (3 

=  {TjyyKq  .  qa  =  (Ti3)"(Tg)-a  =  (Tjjg)-a 

=  Kj3g  ,  pq  .  a   =  Kjjg  .  y 

.•.  K^jg  =  KgKj;, 

• 
which,  by  the  Associative  law,  gives 

Kn  =  n'K (158). 

14.  Show  that  K(-g)  =  - Kg. 

15.  Show  that 

T(p  +  qf  =  (P  +  q)  (Kp  +  Kg) 

=  (Tp)2  +  (Tg)2  +  2S.i7Kg 

=  {T2:>y  +  (Tg)2  +  2 TpTgSU  .  pKq 

=  (Tj)  +  Tg) 2  -  2  TijTg ( 1  - SU  .  jMq) , 

and  therefore  that  T{p-\-q)  cannot  be  greater  than  the  sum  or 
less  than  the  difference  of  Tj?  and  Tg. 

16.  Show  that  gUVg"'  =  TVg  -  SgUVg. 

59.  Applications  to  Plane  Trigonometry. 

1.   For  formulae  involving  2  0,  let 

g  =  Tg(cos2^  +  esin2^). 
Then  .  _ 

Vg  =  g' =  Vlg  (cos  (9  +  e  sin  (9) . 


126  QUATERNIONS. 

From  Equation  (142) ,  S  .  r/  =  (S7)-  +  (ygY,  we  then  have 

Sry  =  (Sr/)2  +  (Yr/)^ 
or,  dividing  out  Tq, 

and,  interpreting, 

cos  26  =  COS"  ^  —  sin^  6. 

Again,  from  Equation  (147),  SU  .  q-  =  2(SU(/)-—  1, 

SU9  =  2(SU^')--1; 
whence 

cos  2^  =  2eos-^  —  1. 

Again,  from  Equation  (142),  V  .  q^=2Sq\q, 

\q  =  2Sq'\q', 

or,  dividing  out  Tq  and  c, 

TYVq  =  2  SUry'TVUfy' ; 
whence 

sin  2  6  =  2eos^  sin^. 

2.   Resmning  Equations  (149)  and  (150), 

TVU^r  =  TVUrySUr  +  SlVyTVUr, 

SVqr  =  SU^SUr  -  TVU^TVUr, 

which  have  already'  been  interpreted  as  the  sine  and  cosine  of 
the  sum  of  two  angles,  and  wi'iting  for 

r  =  T/-(cos<^  +  esin<^),     r~'  =  — (cos<^  —  esin</)), 

q  and  r  being  complanar,  we  have 

TYU^r-^  =  TVlVySU/- -  StVyTYIIr      .     .     (159), 
SUg/-' =  SlVySU?- +  TYU^TYU/-      .     .     (160), 
or,  interpreting, 

sin  {6  —  <^)  =  sin  0  cos  ^  —  sin  <^  cos^, 
cos  (5  —  <^)  =  cos  0  cos  ^  +  sin  ^  sin  ^. 


GEOMETRIC   MULTIPLICATION    AND   DIVISION.        127 

3.  Adding  Equations  (U9)  and  (159), 

TVUgr  +  TYVqi-^  =  2  SUrTVUg, 

in  which,  if  qr  =  p,  qr~^  —  t,   .-.  q  =  V/>f,  r  =  -y/pt'^ (Art.  58,  3), 

T\Vp  +  TYl]t  =  2fil](VpF')TY\]{Vpt)      .     (IGl), 
or 

sinx  +  sin?/  =  2cos|^(a;  —  y)  sin|^(a;  +  y). 

Similarly,  by  subtracting  tlie  same  equations, 

TVUgr  -  TYVqr'^  =  2  SUgTVUr, 

TYVp-TYVt  =  2H\]{-Vpt)TYV{Vpr')  .     (162), 
or 

sin  X  —  sin  y  =2  cos  ^  {x  -\-  y)  sin^  {x  —  y) . 

4.  From  Equations  (150)   and  (160),  by  addition  and  sub- 
traction, we  obtain,  in  a  similar  manner, 

SUi)  +  SUi  =  2SU(ViT0SU(Vy^0     •     •     •     (163), 
and 

SVp  -  SU^  =  -  2  T VU  ( Vpt)  T vu  ( Vpr') , 
whence 

cos  a;  +  cos  y  =  2  cos  ^(x-\-y)  cos  ^(x  —  y), 
cosy  —  cosa;  =  2 sin -1^(0;  +  y)  sin^{x  —  y), 

5.  Resuming  Equation  (152), 


TY  Vg  =  Vi(Tfy-Sg) , 

it  may  be  put  under  the  form 

2(TVUV^)-  =  l-SUg, 
or 

2sin2i^  =  l-cos^. 

and,  in  a  similar  manner,  from  Equation  (151), 


2(SUVg)'  =  SUg+l, 
or 

2cosH^  =  l+cos^. 


128  QUATERNIONS. 

6.  From  Equation  (142) 

(TV:S)./  =  ^MI^^ 

2TVr/  (SrjY 


2  (TV  :  S)7 


1- [(TV:  8)9]^ 
or 

.      o^        2tan^ 

tan  2  ^  = — . 

l-tan^e 

And,  in  a  similar  manner, 

cot-^-1 


cot  2  6  = 


2cot^ 


7.  From  Equations  (90)  and  (91),  q  and  /•  being  complanar, 

Sfyj-  =  SqSr  +  S  .  \q\r  =  SqUr  -  T\qT\r, 
T\qr  =  SqT\r  +  S^-TVr/, 

we  have,  by  division, 

(TV:S)gr  =  5^IX!:+^^:^ 
'^  ^^        SryS/- -  TV7TVy• 

(TV:S)r+(TV:S)9 


or 


Also 


or 


tan(^  +  <^)  = 

(TV:  8)^/-'  = 

tan(^-<^)  = 


1-(TV:  S)y(TV:  S)r 

tan<^  +  tan^ 
1— tan^  tan^ 

(TV:  S)r/-(TV:  S)r 
1  +  (TV:  S)ry(TV:  S)r 

tan  $  —  tan  cf> 


l  +  tan^  tan<^ 
8.   Adding  and  subtracting 

(TV:S)p  =  :^,    (TV:S)<  =  f , 


GEOMETRIC   IVIULTIPLICATION   AND   DIVISION.         129 

we  have 

(TV:S)i9  ±  (TV:S)« 

_  TTpS^  ±  T\7Sp  ^  TYUpSU^  ±  TYOSUp^ 

Hence,  from  Equations  (l-iO)  and  (159), 

(TV  :  S)p  ±  (TY  :  S)  ^  =  "--^'    , 

or 

,  ,   ,  sin  (x  ±  y) 

tan  X  ±  tan  u  =  ^^ ^  • 

cosx  cosy 

By  a  similar  process, 

,       ,        ,  sin  (y  ±  x) 

cot  a.*  ±  cot?/  =  —_ — ■- — : — '-' 
sinx  smy 

9.  From  Equations  (161)  and  (1G3) 

,-      TYUp  +  TYm 

TYUV/^i  = — 7=^-' 

^  9  HJJ-\/nf.-^ 


whence 


or 


2  su  Vi^r 

Jp  +  SI 

2SuVi9r 

TYU«  +  TYU« 
(TVU  :  SU)  yp  =  (TY  :  S)  Vi^^  =        ',....   > 


^     SUiJ  +  su« 

-'  9STTa/)-i^-1 


SUi^  +  SUf 


COSX-  +  COS?/ 


And,  in  a  similar  manner,  from  Equations  (162)  and  (163), 

(TY.b)Vi3«    _    ^^^._^^^^  . 

or 

^      1  .  .        sin  X  —  sin  ?/ 

tani(a;-2/)  = _ ^. 

coscc  +  cos?/ 


130  QUATERNIONS. 

10.  Similar  formulae  ma}'  be  deduced  for  functions  of  other 
ratios  of  an  angle.  Thus,  from  Equation  (90),  writing  rs  for 
?-,  and  making  q  ~r  =  s  all  complanar,  we  have,  by  Equation 
(142), 

s.(/  =  {sqy-3Sq{Ty(jy-, 

or 

cos  36  =  cos'^O  —  3  cos  0  sin- 6, 

or,  under  the  more  familiar  form, 

cos3^  =  4cos='6-3cos^. 


CHAPTER   III. 
Applications   to   Loci. 

60.  Any  vector,  as  p,  may  be  resolved  into  three  component 
vectors  parallel  to  any  three  given  vectors,  as  a,  y8,  y,  no  two 
of  which  are  parallel,  and  which  are  not  parallel  to  an}'  one 
plane.     Thus 

p  =  Xa  +  y[i+Zy (164) 

refers  to  anj-  point  in  space. 

If  the  A'ariable  scalars  x^  y,  z  are  functions  of  two  independ- 
ent variable  scalars,  as  t  and  w,  p  is  the  vector  to  a  surface, 
which,  if  the  functions  are  linear,  will  be  a  plane.     We  ma}', 

therefore,  write 

P  =  <^(^«) (165) 

as  the  general  equation  of  a  surface. 

If  x^  y  and  z  are  functions  of  one  independent  variable  scalar, 
as  ^,  p  is  the  vector  to  a  curve,  which,  if  the  functions  are 
linear,  becomes  a  right  line.     We  may,  therefore,  write 

P  =  <^(0 (166) 

as  the  general  equation  of  a  curve  in  space. 

If  a,  y8,  y  are  complanar,  we  may  replace  either  two  of  the 
vectors  in  Equation  (164)  by  a  single  vector,  in  which  case 
p  =  <^(i)  contains  but  two  variable  scalars,  functions  of  f,  and 
is  the  equation  of  a  plane  curve,  or  of  a  straight  line  if  the  func- 
tions are  linear. 

The  essential  characteristic  of  the  various  equations  of  a 
straight  line  is  that  they  are  linear,  and  involve,  explicitly  or 
implicitly,  one  indeterminate  scalar. 

131 


132  QUATERNIONS. 

61.  Assuming 

p  =  xa  +  yf3,  (a) 

in  which  x  and  y  are  variable  scalars,  functions  of  a  single  vari- 
able and  indoiiondcnt  scalar,  as  t,  as  the  general  form  of  the 
equation  of  a  plane  curve,  by  substituting  in  an}-  particular  case 
the  known  functions  x=  f{t),  y  =f'{t)^  or  x=f"{y),  we  ma}' 
avail  ourselves  of  the  Cartesian  forms  and  apply  to  the  resulting 
function  in  p  the  reasoning  of  the  Quaternion  method. 

For  example,  suppose  a  and  /3  are  unit  vectors  along  the  axis 
and  directrix  of  a  parabola,  the  origin  being  taken  at  the  focus. 
In  this  case  we  have  the  Cartesian  relation 

f=2px+p\  (6) 

or,  substituting  in  (a), 

'2p 

as  the  vector  equation  of  the  parabola. 

Or,  again,  a  and  /8  being  any  given  vectors  parallel  to  a  diam- 
eter and  tangent  at  its  vertex, 

P  =  |'«  +  f/3  (c) 

is  the  vector  equation  of  a  pai'abola,  in  terms  of  a  single  inde- 
pendent scalar  t. 

62.  Let/(.r)  be  any  scalar  function  as,  for  example, 

f{x)  =  x^. 
Then 

d[f(x):\=2xdxr=[f'(x)](lx. 

If,  however,  f((/)  be  a  function  of  a  quaternion  q,  as,  for 
example,  in  the  above  case, 

.  A<j)  =  <n 

then 

/(7  -f  dq)  =  (q  +  dq)-  =  q-  +  qdq  +  dq  .  q  +  (dq)', 
-'-dlf(q)]  =  qdq-]-dq  .  q, 


APPLICATIONS   TO   LOCI.  133 

which  cannot,  however,  be  written  2qdq,  because  of  the  non- 
commutative  character  of  quaternion  multiplication.  We  can- 
not, therefore,  write,  in  general, 

f?[/(5)]  =  [/'(^^)]f?g, 

or  form,  as  usual,  a  differential  coefficient.  Since  vector,  as 
well  as  quaternion,  multiplication  is  non-commutative,  the  same 
is  true  of  the  differentiation  of  a  function  of  a  vector.     Thus,  if 

/(p)=p^ 

and  in  order  to  write  fZ[/(p)]  =  [/'(p)]c?p,  it  would  be  necessary 
to  determine  a  vector  o-,  such  that  adp  =  dp  .  p,  or 

a  =  dp  ,  pdp~^, 

or,  if  e  be  the  versor  of  dp,  since  the  tensors  cancel, 

o-  =  €p€~^ ; 

that  is  (Art.  56,  18),  we  must  have  p,  e  and  o-  complanar,  or 
Veo-  =  Vpe.  Since  complanar  quaternions  are  commutative,  if  q 
and  dq  are  complanar,  or  if  dq  or  dp  is  a  scalar,  this  peculiarit}'" 
of  quaternion  and  vector  differentiation  disappears.  In  this 
case,  dq  and  dp  being  scalars,  f(q)  or  /(p)  are  quaternion  or 
vector  functions  of  scalar  variables,  to  which  the  ordinar}'  rules 
of  differentiation  are  applicable.  In  fact  we  have  onl}'  to  assume 
such  a  function,  as 

p  =  x'a'  +  X"a"  +  X"'a"'  + =  iSo-a  =  <^(0  , 

in  which  a',  a",  a'", are  constants  and  the  onl}'  variables  are 

the  scalar  multipliers,  to  see  that  the  vectors  a',  a",  a'" are 

to  be  treated  as  constants  and  the  usual  rules  of  differentiation 
applied  to  the  scalar  coefficients. 

Such  equations,  then,  as  those  of  the  parabola,  {h)  and  (c), 


134  QUATERNIONS. 

Art.  01,  in  which  a  and  /3  are  given  constant  vectors,  may  be 
differentiated  as  usual.     Thus,  from 


we  have 


p  =  ^'a+^/?. 


(It 


p  and  p'  being  an}-  two  vectors  to  the  cun-e, 

p'— p  =  Ap 

is  the  vector  secant ;  so  that  wlien  p  and  p'  become  consecutive, 
and  the  secant  a  tangent, 

dp  =  {ta+/3)dt 

is  a  vector  along  the  tangent  to  the  curve  at  the  point  corre- 
sponding to  t.     The  vector  to  this  point  being  -a  +  tfi,  and  x 

any  variable  scalar,  we  :nay  write  the  equation  of  the  tangent 
line  at  that  point 

p  =  *-a-\-tf3  +  x{ta  +  f3)  ; 

for  an}'  given  point,  x  being  the  onl}-  scalar  variable. 

63.  It  has  been  seen  that  the  usual  definition  of  differential 
coeflieicnts  is  inai)plicable  to  quaternions  in  general,  for  this 
definition  involves  the  commutative  property-  of  multiplication, 
which  is  not,  in  general,  true  of  quaternions,  nor  of  the  vectors 
to  which  thoy  may  degrade.  It  becomes  necessar}',  therefore,  to 
give  a  definition  of  differentials  which  shall  not  involve  this  prop- 
ert}-,  3-et  which  shall  also  be  true  of  quaternions  which  degrade 
to  scalars,  and  therefore  be  equally  applicable  to  ordinary  scalar 
quantities. 

If  j)=/(^),  such  a  definition  is  involved  in  the  formula 

c?J^  =  Ji'^i«[/(^  +  H-^cZg)-/(^^)]    .     .     (167), 


APPLICATIONS   TO   LOCI.  135 

for,  let  f(q,  r,  s,  )  =  0  be  an}-  relation  between  a  s^'stem  of 

quaternions  q,  r,  s,  ,  and  let  Ag,  Ar,  As,  be  finite  and 

simultaneous  differences,  so  that  q-\-Aq,  r  +  A?-,  s  +  As,  

satisfy  the  relation /(g,  ?',  s,  )  =  0.    Then  in  passing  from  the 

new  system  g  +  Ag,  to  the  old  system  g,  ,  the  simul- 
taneous differences  can  all  be  made  to  approach  zero  together, 
since  they  all  vanish  together.  If,  while  these  differences  Ag, 
Ar,  thus  decrease  indefinitely  together,  they  be  all  multi- 
plied by  the  same  increasing  number,  ?i,  the  equimultiples  wAg, 

nAr,  ma}'  tend  to  finite  limits,  and  these  limits  are  defined 

to  be  the  simultaneous  differentials  of  the  related  quaternions  g, 
r,  s,  ,  and  are  written  clq,  clr,  ds, Simultaneous  differ- 
entials are,  therefore,  the  limits  of  equimultiples  of  simultaneous 
decreasing  diflTerences.  If,  then,  in  Ap=  f{q+ Aq)— f{q), 
while  the  finite  differences  A^;,  Ag  be  indefinitely  decreased,  they 
be  multiplied  b}'  a  number,  n,  ultimately'  to  be  made  iufinit}^, 
so  that 

9iAi>  =  n  [/(g  +  Ag) -/(g)], 

and  we  pass  to  the  limit,  writing  dp  for  nAp^  and  dq  for  ?iAg, 
we  have 


7         limit 


f[q  +  '^]-f{q) 


a  formula  for  the  differential  of  a  single  explicit  function  of  a 
single  variable. 

ltQ=F{q,r,  ), 

clQ  =l!=l^  niF(q  +  n-hlq,  r  +  n-hlr, )-F{q,  r, )]    (1G8). 

In  these  formulae,  dq,  dr,  are  any  assumed  variables,  no 

reference  having  been  made  to  their  magnitudes,  and  n  any 
positive  whole  number  conceived  so  as  to  tend  to  infinit}'.  To 
show  that  these  differentials  need  not  be  small,  as  also  the  ap- 
plication of  the  formula  to  the  differentiation  of  ordinary-  scalar 
quantities,  let 


136 

then 

whence,  as  usual, 


QUATERNIONS. 


(l/  +  Ay)  =  {x-{-Axy; 


Ay=2xAx  +  {Axy, 

or,  n  being  a  positive  whole  number, 

n  Ay  =2x11  Ax  -\-  n~^{n  A  x)-. 

If,  now,  the  differences  A  y  and  A  x  tend  together  to  zero, 
while  n  increases  and  tends  to  infinit}'  in  such  a  manner  that 
nAx  tends  to  some  finite  limit,  as  a,  we  have,  for  the  other 
equimultiple  n  A  ?/, 

n  Ay  =  '2  xa  +  n~^  a-. 

But,  since  «,  and  therefore  a^,  is  finite,  n'^cr  tends  to  zero, 
and,  at  the  limit,  nAy  =  2xa.  Hence  the  limits  of  the  equi- 
multiples nAx  and  nAy  are  respectively  a  and  2xa,  and 
'dx  =  a,  dy  =  2xa  hy  definition;   from  which 

dy  =  2  xdx. 

For  a  vector  function  we  should  write 

limit 


'^p = ,;= 'i ''  ^-^^p + "~'^^)  -•^(^)] 


and  for  a  scalar  function,  p  =  <^  (<) , 


(1G9), 


(170), 


in  which  latter  t  and  dt  are  independent  and  arbitrar}^  scalars. 
64.   As  a  further  illustration  of  the  definition,  let 

p=cj>{t) 


APPLICATIONS    TO   LOCI. 


137 


Fig.  G4. 


be  the  equation  of  any  plane  curve  in  space,  and  op  =  p  (Fig.  G4) 
a  vector  from  the  origin  to  a  point  p 
of  the  curve  ;  t  being  an}'  arbitrary  sca- 
lar representing  time,  for  example  ;  so 
that  its  value,  for  an}'  other  point  p'  of 
the  curve,  represents  the  interval 
elapsed  from  an}-  definite  epoch  to  the 
time  when  the  point  generating  the 
curve  has  reached  p.' 

If  p'  be  the  vector  to  p'  then 


p'~  P  =  pp'=  Ap 

is  strictly  the  finite  difference  between  p  and  p',  and,  if  the  corre- 
sponding change  in  the,  At, 

pp'=  (p  +  A  p)  -  p  =  A  p=  c^(;  +  A  0  -  <^(0  =  A  c/,(0  ; 

where  op'=  ^(^  +  A  ^) ,  and  A  Hs  the  interval  from  p  to  p! 

In  ^A  i,  p  would  have  reached  some  point  as  p",  for  which 
Op"=  <^(i  4-i^A  ^),  on  the  supposition  that  pp"  is  described  in 
■|  A  i.  On  the  basis  of  this  closer  approximation  to  the  velocity 
at  p,  p  would  have  been  found  at  p",  had  this  velocity  remained 
unchanged,  such  that 

P2y'=  2  pp"=  2(op"-  Op)  =  2[(/>(«  +  A-A  0  -  c/.(0]. 

For  a  closer  approximation  to  the  vector  described  in  A  ^  with 
the  velocity  at  p,  suppose  at  the  end  of  ^At  the  point  is  at  p'", 
for  which  op"'=  (^(i +  |-Ai).  Under  this  supposition,  the  vec- 
tor described  in  A  t  would  have  been 

Pp"'=  3pp"'=  3 (op"'-  Op)  =  3[<^(<  +  iA  0  -  <A(0]. 


and,  at  the  limit,  representing  the  multiple  of  the  diminishing 
chord  by  cZp, 


7         hmit 
dp  =  n 


«+fV^(o 


138  QUATERNIONS. 

65.   Resuming  Equation  (167), 

dp  =  df(q)  =  ^ff  ^n  [fig  +  n-'dq)  -/(?)],  (a) , 

the  second  member  ma}-  be  written  f{q^  dq) ,  but  not,  as  ordi- 
narily, /(g)  d^. 

In  /(</,  dq),  dq  ma}'  be  composed  of  parts,  as  q',  q",  q'", , 

with  reference  to  which /(g,  dq)=f{q,  q'+q"+ )  is  distrib- 
utive.    To  prove  this,  let 

dq  =  q'+q"; 
we  are  to  prove  that 

f{q,q'+q")=f{q,q')+Aq,q"). 

Since  before  passing  to  the  limit,  the  second  member  of  (a) 
is  a  function  of  n,  q  and  dq,  we  ma}'  express  this  function  by 
the  symbol  /„(g,  dq),  and  write 

f{q,  dq)  =  n[f{q  +  7r'dq)  -f{q)^=Mq,  dq), 
or 

f(q  +  n-hlq)  =f{q)  +  ir'Uq.  dq) . 
Replacing  dq  by  q'  and  q"  in  succession,  we  have 

f{q  +  n-'q')  =f{q)  +  n-\Uq-  Q') , 
f{q+n-'q")=f{q)+n-'Mq,q"), 

and,  following  the  same  law  of  derivation, 

f(q  +  n-'q"+n-'q')=f(q  +  n-'q")  +  n-'f,Xq  +  n-'q",q'), 
f{q  +  n-'q'+n-'q")=f{q)  +  n-'Uq,q'+q"), 

from  which 

/„(^/,  q'+  q")  =UQ'  q")  +fM  +  n-'q",  q'), 

the  limiting  form  of  which,  for  7i  =  x,  is 

/(9,9'+^")=/(7,0+/(9,5')       .     •     (171), 


APPLICATIONS   TO   LOCI.  139 

which  may,  in  like  manner,  be  extended  to  the  case  of 

dq  =  q'+q"+q"'+ 

It  follows  from  the  above  that,  ^f^  p=f{q,  xclq), 

f(q,xclq)  =  xf{q,dq)      .     .     .     .     (172). 
If  Q  =  F{q,  r, ),  whence.  Equation  (168), 

dQ  =  d[F(q,r, )] 

=  ,f  ™'^  n  [_F{q  +  n-'  dq,  r  +  n'^ dr, )  -  F(q,  r. )  ] , 

the  last  member  wiU  be  a  linear  and  homogeneous  function  of 

dq,  dr,  ,  and  distributive   with  reference  to  each  of  them. 

Hence,  to  differentiate  such  a  function,  we  do  so  with  reference 
to  each  factor,  and  take  the  sum  of  the  results  obtained,  as  usual ; 
taking  care,  however,  not  to  make  use  of  the  commutative  prop- 
erty-.    Thus  d{qr)  =  dq  .  r  +  qdr,  but  not  rdq  +  qdr. 

66.  When  g  is  a  function  of  anj'  variable  scalar  t^  represent- 
ing time,  for  example,  then,  if  t  be  given  a  finite  increment  A  t, 
for  which  the  corresponding  one  of  g  is  A  g,  we  have 

Aq  =  Aiv  +  Axi  +  A  yj  -\-Azk; 

and,  if  the  several  paints  of  the  quaternion  vaiy  continuously 
with  the  independent  variable  t,  at  the  limit  we  may  form,  as 
usual,  the  differential  coefficient 

dq     dw     dx .  ,  dy  .  ,  dz, 

—  = 1  -1 '—  1  -\ K. 

dt       dt       dt         dr       dt 

The  successive  differential  coefficients,  as  also  the  partial  ones, 

when  q  =  (fi[t,  v, ) ,  are  derived  from  the  quadrinomial  form  in 

the  same  manner. 


140  QUATERNIONS. 

67.   Examples. 
1.    To  find  clTq. 

dTq  _  rZVtt^  +  x^  +  y-  +  z- 
dt  ~  iU 

1  /    dv}  ,     dx  ,      dif  ,     dz\ 
Tq\     dt  dt         (It        dtj 

_±«      'hizn-^     dq  VKrjTq 
-Tq^'dt*^^-^'dt   ~Tf~ 
dq 

dt   VqTq  q 

dq 


or 


dlq  _f.fU_ 
"dr~    Vq' 

2.  (Tpy-  =  -p\ 

The  first  member  being  a  scalar,  we  have 

2TpdTp. 

From  the  second  member 

J,  o,       limit      r/     .      -lis"        on 
f?(p-)  =„  ^  -^  "  l(p  +  n-'dp)-  -  p2] 

=  limit  pdp  +  dp  ,  p  +  n~^{dp) 
=  pdp  +  dp  .  p  =  '2  Hpdp. 
Equating 

TpdTp  =  -  Hpdp. 

From  this  we  may  obtain 

dTp  =  -  S  .  Vpdp  =  S^^ 

dTp_    dp 
Tp-%- 

3.  To  find  dVq.     We  have 

TqVq = q ; 
dTq  .  Vq  +  CZU7  .  1q  =  dq, 


whence 


APPLICATIONS   TO   LOCI.  141 


clTq  .  JJq      cWq  .  Tg  _  dq 
HqUq  Tr/Ug     ~  g' 


dUg  _  dq  _  dTg 

Ug        g        Tg ' 


and,  substituting  from  Ex.  2, 


Ug       g       '   g  ' 
c?Ug  _  Y  f?g 
Ug  q 

or 

dUg  =  Y^  .  Ug. 

q 

4.    From  the  above  expressions  for  fZTg  and  cZUg,  we  have 

dq  =  dlq  .  Ug  +  TgdUg 
^/^£?g         rfg\ 
V    Ug         Ug; 


fy        q. 

as  the  form  under  which  the  differential  of  a  quaternion  may 
alwa3-s  be  -uTitten. 

5.    To  find  dUp.     We  have,  from  p  =  TpUp, 

dp  =  dTp  .  U/3  +  TpdVp, 
dp  __  dTp      (TVp 
p"  Tp        Up 

=  S^  +  '-^,  from  Ex.  2, 

P        Up' 

or 

dVp  ^dp  _^^  dp  _  y  ^  _  y  ^?P  ♦  P   ^  Jpdp^    _   g|.^,_  ^ 

up        p  p  p  p"  (Tp)- 


whence,  also, 

_  pT  .  r?pp 

(Tp)^    ■ 


<^^P  =  -     .™,    ^3 


142  QUATERNIONS. 

6.  From  the  above  expressions  for  clTp  and  r7Up, 

dp  =  dip  .  Up  +  p^^-^^. 

7.  That  S,  V  and  K  are  commutative  with  d  is  seen  from  the 
following : 

whence 

dq  =  d^q  +  d\q,  (a) 

and,  since  dq  is  a  quaternion, 

dq  =  Srfry  +  Jdq,  (6) 

hence 

dSg'  =  Mq     and     cZTg  =  Ydq.  (c) 

Again 

whence 

dKq  =  dHq  —  d\q, 

and,  taking  the  conjugate  of  dq  in  either  (h)  or  (o),  we  have, 
with  or  without  (c) , 

dKq  =  Kdq. 

8.  (Try)2  =  ryKg. 

2T5(my  =  ^J*™^  7i  [(^  +  n-'d5)  (Kg  +  n-^ZKry)  -  gKg] 

=  limit  [dq{Kq  +  jr^  Kdq)  +  gKcZg] 

=  fZ^  .  Kq  +  ryKrZ(7 

=  K  .  qKdq  -f-  ryKfZg 

=  2  S  .  qKdq  =  2  S  .  K(?cZg,  [Equation  (80)] 

or,  since  Tq  =  TKg  and  UK7  =  U  -  =  — , 

q      Vq 

dTq  =  S  .  U  -  cZry  =  S  .  Vq''  dq. 

If  g  =  a  vector,  as  p,  then,  since  Kp  =  —  p,  this  becomes 

cZTp  =  -  S  .  Updp, 
as  in  Ex.  2. 


APPLICATIONS   TO    LOCI.  143 

9.  r—(^. 

dr  =^  "^^  n  [(g  +  n-^  dqY  -  r/] 

=  limit  [gdg  +  dg  .  g  +  ?i~^(dg)-] 
=  gdg  +  clq  .  g  ; 
.-.  dr=2  ^qdq  +  2  SgTdg  +  2  SdgVg. 

If  g  =  a  vector,  as  p,  then  Sg  =  0,  ScZg  =  0,  aucl 
d{p^)=2Spdp 
as  in  Ex.  2. 

10.  r=  Vg.     Then  q  =  'i",  and,  as  before, 

dq  =  rdr  +  cZr  .  r. 

Operating  with  ?'  X  and  X  Kr,  in  succession, 

rdq  =  r~  dr  +  irh  .  r, 
(Zg  .  Kr  =  rd7'  .  Kr  +  dr  .  7-Kr 
=  rdr  .  Kr  +  {Trydr, 
or,  adding, 

rdq  +  dg  .  Kr  =  [r-  +  (Tr)-]r?r  +  rdr{r  +  Kr) 
=  [r'  +  (Ti')'  +  2Sr  .rJcZr, 

which  gives  dr  =  d\/q  in  terms  of  dq. 

1 1 .  gg-i  =  1 .     "We  have 

qd{q-^)-\-dq  .  g-^  =  0. 
Operating  with  g"^  X 

g-^gcZ(g-i)  +  g-^f?g.g-^  =  0, 

d-  = dq  .  -• 

q  <1  <1 


144  QUATEKNIONS. 

If  ^  =  a  vector,  as  p, 

a-  = dp- 

P         P     P 

1,1,1,       11, 

= dp-  +  —dp -dp 

p      p      p-  p  p 

P'        P  \P  PJ 

^dp_2^dp 


P       P     P 

P  PJP 


=  f^^_2S^V  =  -K^^.l. 


12.    Differentiate  SUg. 


c?SU5  =  MVq  =  S  .  Y  ^  Vq       [Exs .  7  and  3 . ] 

da 
=  S.— VUg 

=  S  .^UVfyTYU^ 

=  -S.  J^TVUg. 
q\J\q 

13.  Differentiate  TU^. 

dq 
dTVq  =  T .  fZU7  =  V .  V—  Ug        [Exs.  7  and  3.] 

=  Y.Vq-'Y{dq  .  q-'). 

14.  Differentiate  TWq. 

tZTVUg  =  s!^^^3^  [Ex.  2.] 

^  V\q  "-  -' 

dq 

7TT  V— Ug 

~^V\q      '     V\q 

dq 
=  ^'qljrq^^^' 


APPLICATIONS    TO    LOCI.  145 


The  Right  Line. 

As  in  Cartesian  coordinates,  the  form  of  the  equations  of  a 
right  line,  as  of  otiaer  loci,  will  depend  upon  the  assumed  con- 
stants, and  in  an}^  given  problem  one  form  may  be  more  con- 
veniently used  than  another. 

68.   Right  line  through  the  origin. 

If  o  be  the  initial  point,  or  origin,  and  p  =  or  a  variable  vec- 
tor in  the  prolongation  of  a  =  oa,  then 


(173) 


is  the  equation  of  a  right  line  through  the  origin  in  the  direction 
of  the  constant  vector  a. 
The  equations 

^P  =  ^<^\ (174) 

cbviousl}'  refer  to  the  same  right  line. 

Since  any  line,  represented  as  a  vector  by  a,  is  parallel  to 
p  =xa,  we  may  say  that  the  above  equations  are  those  of  a  right 
line  through  the  origin  parallel  to  a  given  line  ;  or,  a  being  a 
point  given  by  a  =  oa,  they  are  the  equations  of  a  right  line 
through  the  origin  and  a  given  point. 

69.   Parallel  lines. 

If /3  =  OB  be  a  constant  vector  to  a  given  point  b,  then 

p  =  (3  +  xa (I'^S) 

is  the  equation  of  a  right  line  through  a  given  point,  and  parallel 
to  a  given  line,  as  p'=  xa  through  the  origin.  Or,  a  being  a  giA-en 
vector,  it  is  the  equation  of  a  right  line  through  a  given  point 
and  having  a  given  direction.  If  a  is  an  undetermined  A-ector, 
it  becomes  the  general  equation  of  any  one  of  the  infinite  num- 
ber of  right  lines  which  may  be  drawn  through  a  given  point.  If 
0  and  B  coincide,  /3  =  0,  and,  as  before,  p  =  xa. 


146 


QUATERNIONS. 


a  remaining  the  same,  and  ^'  =  ob'  being  a  vectoi-  to  any  other 
point  l^',  for  the  equations  of  two  parallels,  we  have 


p  =  /3  +xa   I 
p  =  /3'+x'a) 

or,  since  a  and  p—jS  are  parallel, 


(176), 


'^<f>-f')  =  n (177). 

70.   Right  line  through  two  given  points. 

If  0A  =  a    (Fig.   65),  OB  =  /3   are   the  vectors  to  the  given 
points,  and  p  the  variable  vector  to  an}- 
Fi^.  65.  point  R  of  the  line  whose  equation  is  re- 

R    quired,  we  have 


and 


AR  =  x\n  =  x{ft  —  a), 

OR  =  OA  +  AR, 

or,  for  the  required  equation, 

p  =  a+.r(/3  — a) 


(178), 


which,  if  one  of  the  points,  as  a,  coincides  with  the  origin, 
becomes  p  =  x^,  as  before. 

"We  have  seen,  Art.  55,  that  if  Sa^y  =  0,  a,  /3  and  y  are  com- 
planar.     Replacing  y  by  the  varia1)le  vector  p, 

Saf3p=0 (179) 

is  tJie  equation  of  a  plane.,  since  it  expresses  the  condition  that  p 
is  complanar  with  a  and  /?.  If  we  have  also  Sayp  =  0,  the  two 
equations,  taken  together,  represent  the  line  of  intersection  of 
these  two  planes. 

These  equations  may  be  obtained  from  the  line  p  =  xa  bv  ope- 
rating with  S(Vay8)  X  and  S(Vay)x  ;  or,  conversel}-,  to  find  the 
equation  of  the  line  in  terms  of  known  quantities,  having  given 

Sa/3p  =  0,      Sayp  =  0, 


APPLICATIONS   TO   LOCI.  147 

write  these  latter  under  the  form 

S.pya^  =  0,      S.pVay=0, 

whence  it  appears  that  p  is  perpendicular  to  both  Ya/3  and  Yay, 
and  is  consequeutl}'  parallel  to  the  axis  of  their  product ; 
therefore  "^ 

p  =  yY  .  Ta/3Yay 
=  l/(ySa/?a-aSa^y)  [Eq.  (112)] 

=  —  2/aSa^y, 

or,  putting  —  ySajSy  =  x, 

p  =  Xa. 

71.   Right  line  perpendicular  to  a  given  line. 

1.  Let  8  =  CD  (Fig.  Q6)  be  a  vector  through  the  origin.  To 
find  the  equation  of  dc  through  its  extremity  -pig.  66. 

and  perpendicular  to  it.      Now  p  —8  is  a  d  k      c 

vector  along  dr,  and  therefore  b}'  condition 

SS(p-8)  =  0. 

Whence  SSp  =  -(T8)-,  or 

^Sp  =  c,  a  constant (180). 

In  order  that  p,  p  —  8  and  S  be  complanar,  we  must  have 

S.Sp(p-8)  =  0, 
or 

S  .  (V8p)  (p  -  S)  =  0. 

2.  p  —  8,  being  perpendicular  to  both  8  and  YSp,  will  be 
parallel  to  the  axis  of  their  product,  or  to  V  .  8Y8p.  Hence,  if 
y  =  GO  be  a  vector  to  any  point  c,  in  the  plane  of  od  and  dr,  the 
equation  of  a  right  line  through  a  given  point  c,  perpendicular  to 
a  given  line  od,  wUl  be 

p  =  y  +  a;V.8VSy (181). 


148  QUATERNIONS. 

3.  If  the  perpendicular  is  to  pass  through  the  origin,  then, 
from  Equation  (180), 

SSp  =  0 (182), 

or,  in  another  form,  from  Equation  (181),  y  being  parallel  to 

V  .  8V3y, 

p  =  y\.8\8y (183). 

4.  The  student  will  find  it  useful  to  translate  the  Quaternion 
into  the  Cartesian  forms.   Thus,  from  Equation  (180) ,  if  rod=  6, 

S8/3  =  -T8Tpcos^, 

whence,  if  r  and  d  represent  the  tensors, 

rd  cos  6  =  d-,     or     ?•  = 


cos  6^ 
the  polar  equation  of  a  right  line. 

5.  Equation  (181),  of  a  line  through  a  given  point  and  per- 
pendicular to  a  given  line  through  the  origin,  ma}'  be  otherwise 
obtained,  as  follows : 

Let  y  and  8,  as  before,  be  vectors  to  the  point  and  along  the 
given  line,  respectiveh',  and  ft  a  vector  along  the  required  per- 
pendicular, whose  equation  will  then  be 

p  =  y  +  a-/?.  (a) 

To  eliminate  /S  we  have  the  conditions 

S8(3  =  0, 

since  8  and  (3  are  perpendicular  to  each  other,  and 

Sy8)8  =  0, 

since  y,  8  and  /5  are  complanar.  But  T8y  is  perpendicular  to  this 
plane,  and  therefore  V  .  SV8y  is  parallel  to  /3 ;  hence,  substitut- 
ing in  (a), 

p  =  y  +  x\  ,  8V8y, 
or  simpl}^ 

p  =  y  +  x6\oy. 


APPLICATIONS   TO   LOCI.  149 

K  SyS/?^  0,  y,  S  and  /S  are  not  complanar,  and  the  problem  is 
indeterminate  ;  which  also  appears  from  (a) ,  b}'  operating  with 
X  S  .  8,  whence,  since  S/S8  =  0, 

Sp8  =  Sy8, 

a  result  which  is  independent  of  /3.  and  an  infinite  number  of 
lines  satisf\'  the  condition. 

6.  If  the  line  to  which  the  perpendicular  is  ch-awn  does  not 
pass  through  the  origin,  let 

P  =  f3-i-xa  (a) 

be  its  equation.     Then,  if  p  be  the  vector  to  the  foot  of  the  per- 
pendicular, we  have  Sa(p  —  y)  =  0,  or 

Sa(.ra  +  iS-y)  =  0,  (6) 

because  the  line  is  peq^endicular  to  (a) ,  or  its  parallel  a.   Hence, 
from  (&), 

Xa  =  a~^  Sa(y  —  y8)  , 

or,  for  the  perpendicular  p  —  y, 

p  -  y  =  .ra  +  /5  -  y  =  a'^ Sa(y  -  /5)  -  a-^a(y  -  /?) 
^_a-iYa(y-/?). 

Its  length  is  evidently 

TT[tJa.(y-/3)] (184). 

7.  This  perpendicular  is  the  shortest  distance  from  the  point 
to  the  line.  The  problem  may,  therefore,  be  stated  thus  :  to 
find  the  shortest  distance  from  c  to  the  line  p  =  xa-\-  (3.  p  being 
the  vector  from  c  to  any  point  of  the  given  liue.  this  vector  is 

(3  +  Xa  —  y, 

and,  in  order  that  its  length  be  a  minimum, 

dT(^  +  a;a-y)  =  0 
=  T(^  +  xa  -  y)dT(l3  +  xa  -  y) 
=  -  S[  (/8  +  Xa  -  y)  a]dx  =  0, 


150  QUATERNIONS. 

or 

S(/?  +  a-a-y)a  =  0, 

that  is,  the  line  must  be  perpendicular  to  p  =  .ra  +  /?. 

8.  If  the  perpendicular  distance  from  the  origin  to  p  =  (3  +  xa 
is  required,  p,  being  as  before  the  vector  to  the  foot  of  the  per- 
pendicular, coincides  with  it ;  hence,  y  being  zero,  and  8  repre- 
senting this  value  of  p, 

8  =  xa  +  ^. 

Operating  with  X  S  .  8,  since  SaS  =  0, 

-  {Td)- =  i>(38. 


Hence 


or 


^•>     S,'3S      S.^T8U8 
1 J  —  — —  — , 

T8            T8 
T3  =  S./3US (185). 


72.  "We  are  to  observe  that  the  foregoing  equations  of  a  right 
line  are,  as  remarked  in  Art.  GO,  all  linear  functions  invqjving, 
explicitly'  or  implicitly,  a  single  real  and  independent  variable 
scalar.     Such  is  evidently  the  case  for  such  equations  as 

p  =  xa,  [Eq.  (173)] 

p=:/3  +  xa,  [Eq.  (175)] 

p  =  a-\-x{(3-a).  [Eq.  (178)] 

So  also  for  the  implicit  forms,  as  Vap  =  0  [Eq.  (174)]  ;  em- 
ploying the  trinomial  forms 

a  =  ai  -\-hJ  +  ck, 
p  =  xi  +  >/J  +  zk, 
we  have 

ap  =  (bz  —  cy)  i  +  {ex  —  az)j  +  (cuj  —  bx) k  —  (ax  +  bij  -\-  cz) . 

Whence 

Va/j  =  (bz  —  cy)  i  +  {ex  —  az)j  +  {ay  —  bx)  k  =  0; 
.-.  bz  =  ey,     cx  =  az^     ay  =  bx, 

in  which  x  and  y  are  functions  of  z. 


APPLICATIONS   TO   LOCI. 


151 


The  Plane. 

73.  Equation  of  a  plane. 

1.  If,  in  the  equation  S  .  8/3  =  0,  ■which  denotes  that  /3  is  per- 
pendicular to  8,  we  replace  /3  b}-  tlie  variable  vector  p, 

S  .  8p  =  0 (186) 

is  the  equation  of  a  plane  througli  the  origin  perpendicular  to  8. 

2.  Or,  let  8  =  CD  (Fig.  66)  be  the  vector-  Fig.  66  (6is). 

perpendicular  on  the  plane,  and  dr  any  line       d r      c 

of  the  plane.  /     -^ 

Then 


or 


SS(p-S)  =  0, 
SS/)  =  8-  =  -(TS)2, 

SSp  =  c,  a  constant 


(187) 


is  the  general  equation  of  a  plane  perpendicular  to  S.     Here  dr 
is  any  line  of  the  plane  ;  and,  if  \8p  =  e. 


Sep  =  an  indeterminate  quantity 


(188), 


If  the  plane  pass  througli  the  origin,  we  have,  as  liefore, 
SSp  =  0.  Conversel}',  if  SSp  =  c  is  the  equation  of  a  plane,  8  is  a 
vector  perpendicular  to  the  plane. 

3.  The  equation  of  a  plane  through  the  origin  perpendicular 
to  8  may  also  be  written  in  terms  of  an}'  two  of  its  vectors,  as 
y  and  (3 ; 

P  =  xf3  +  yy. 

Both  of  these  indeterminate  vectors  may  be  eUminated  by 
operating  with  S  .  8  X  ,  whence 

SSp  =  0 

as  before  ;  or  one  may  be  eliminated  by  operating  with  V  .  )8  X  , 
whence 

T/Sp  =  z8, 


162  QTJATERNIOXS. 

from   which   we   may   again  derive    SSp  =  0  In*  operating  with 
Y  .  8  X  ;  for 

V  .  STySp  =  \z^-  =  0 

=  pSS/3-/3SSp,  [Eq.  (Ill)] 

whence,  since  SS^  =  0,  SSp  =  0. 

4.  The  equation  of  a  plane  through  a  point  b,  for  which 
OB  =  ;8,  and  perpendicular  to  8,  is 

S8(p-^)  =  0 (189). 

5.  Having  the  equation  of  a  plane,  SSp  =  c,  to  find  its  dis- 
tance from  the  origin,  or  the  length  of  p  when  it  coincides  with 
8,  we  have  p  =  xS  ;  hence 

S8/3  =  c  =  SxS-  =  xB^, 
or 


which,  in  p  =  a;8,  gives 


or 


x  = 

c 

p  = 

C 

Tp  = 

c 

"t8 

(190). 


74.    To  find  the  equation  of  a  plane  through  the  origin,  making 
equal  angles  with  three  given  lines. 

Let  a,  ft,  y  be  unit  vectors  along  the  lines.     The  equation  of 
the  plane  will  be  of  the  form 

S8p  =  0. 

By   condition,    Sa8  =  S^8  =  SyS  =  T8  sin  <^  =  x,    4>  ^ing   the 
common  angle  made  b}'  the  lines  with  the  plane. 

Hence 

.     ,        X 
sm  d)  =  — -. 
^      TS 


APPLICATIONS   TO   LOCI.  153 

To  eliminate  S,  we  have,  from  Equation  (118), 

8Sa(3y  =  YaySSyS  +  T/SyHaS  +  TyaS/88, 

and,  by  condition, 

8Sa/5y  =  X  (VafS  +  \/3y  +  Tya)  . 

Tlie  vector  represented  by  the  parenthesis  is,  then,  the  per- 
pendicular on  the  plane,  whose  equation,  therefore,  is 

Sp(Ya/3  +  y/3y  +  yya)=0  .      .      .      .      (191), 

and  the  sine  of  the  angle  <^  is 

So^y 

T(ya;8+Vy8y  +  Vya)' 

75.  Equation  of  a  plane  through  three  given  points. 

Let  a,  /?,  y  be  vectors  to  the  given  points  ;  then  are  the  lines 
joining  these  points,  as  (a  —  ^),  (/S  —  y),  lines  of  the  plane.  If 
p  is  the  variable  vector  to  any  point  of  the  plane,  p  —  a  is  also  a 
line  of  the  plane.     Hence 

S(p-a)(a-^)(^-y)  =  0, 

or 

S(pa^  -  pay  -  p/5-  +  p^y  -  a"/?  +  a'y  +  a/^"  -  a^Sy)  =  0. 

But 

S(-p^2)  =  0,  S(-a2/3)  =  0,  etc., 

S(—  pay)  =  Spya  =  S  .  pVya, 
Spa^  =  S  .  pTa^,  etc. , 

lienc6 

S.p(Va^  +  T/Sy  +  Vya)-Sa/3y=0    .      .      (192), 

which,  by  making  the  vector-parenthesis  =  8,  may  be  written 
under  the  form 

Sp8  -  Sa^y  =  0, 


154  QUATERNIONS. 

in  which  8  is  along  the  perpendicular  from  the  origin  on  the 
phme.  "When  p  coincides  with  this  perpendicular,  p  =  x8,  and, 
from  the  above  equation, 

x8-  =  SaySy, 

or,  for  the  vector-perpendicular, 

p  =  .T8  =  8-iSa/Sy=  ^'^^V 


Va^  +  V^y+Vya 

76.  "We  observe  again,  from  inspection  of  the  equations  of  a 
plane,  that,  as  remarked  m  Art.  60,  the}'  are  linear  and  func- 
tions of  two  indetermmate  scalars.  Thus,  for  a  plane  through 
the  origin 

SSp  =  0,  [Eq.  (186)] 

emplo3ing  the  trinomial  forms  8=ai-\-bJ-\-c]c  and  p=xi-\-i/j-{-zk, 
we  obtain 

8p  =  (bz  -  cy)  i  +  {ex  -  az)j  +  {ay  —  bx)k  —  {ax  +  by  +  cz) , 

the  last  term  of  which  is  the  scalar  part ;  hence 

ax  +  hy  -f  cz  =  0, 

the  equation  of  a  plane  through  the  origin  o,  perpendicular  to  a 
Inie  from  o  to  (a,  6,  c),  which  may  be  written /(x,  y,  z)  =  0  ; 
or  as  a  function  of  two  indetormiiuites.  In  the  same  way,  from 
an  inspection  of  the  other  forms, 

p  =  xa  +  yji,  [Art.  73,3] 

P  =  8  +  .ra  +  2//3, 
SS/3  -  c'  =  ax  +  hy  +  cz-c'  =  0,         [Eq.  (1«7)] 

we  observe  the)'  are  linear  functions  of  two  indeterminate  scalars. 

77.  Exercises  and.  Problems  on  the  Right  Line  and 
Plane. 

1.  /8  and  y  being  vectors  along  tico  given  lines  icJiich  intersect 
at  the  point  a,  to  ivhich  the  vector  is  oa  =  a,  to  ivrite  the  equation 
of  a  line  perpendicular  to  eaph  of  the  two  given  lines  at  their 
intersection. 


APPLICATIONS   TO   LOCI.  155 

T/3y  is  a  vector  in  the  direction  of  the  required  hne,  whose 
equation,  therefore,  is 

p  =  a  +  x\(Sy (193). 

If  a  =  oa'  be  a  vector  to  any  other  point,  as  a^  then  is 

p  =  a'  -\-  X\^y 

the  equation  of  a  hne  through  a  given  point  perpendicular  to  a 
given  plane  ;  the  latter  being  given  by  two  of  its  hnes. 

2.  a  and  (3  being  vectors  to  two  given  points,  a  and  b,  and 
S8p  =  c  the  equation  of  a  given  ■plane,  to  fiyid  the  equation  of  a 
plane  through  a  and  b  perpendicular  to  the  given  plane. 

8,  p  —  a  and  a  —  /?  are  lines  of  the  required  plane,  hence 

S(p-a)(a-/3)8  =  0, 
or 

Sp(a-/3)S  +  Sa/3S  =  0 (194) 

is  the  required  equation. 

3.  oc  =  y  being  a  vector  to  a  given  p>oint  c,  and  p  =  a-\- x/3, 
p  =  a'-\-x'fS'  the  equations  oftioo  given  lines,  to  ivrite  the  equation 
of  a  jilane  through  c  parallel  to  the  two  given  lines. 

If  lines  be  drawn  through  the  given  point  parallel  to  the  given 
lines,  they  will  lie  in  the  required  plane.  As  vectors,  /5  and  /3' 
are  such  lines,  and  p  —  y  is  also  a  line  of  the  plane.     Hence 

S^^'(p-y)=0 (195) 

is  the  requii'ed  equation.  If  y  =  a,  or  a',  it  is  the  equation  of  a 
plane  through  one  line  parallel  to  the  other.  Or,  if  y  is  inde- 
terminate, it  is  the  general  equation  of  a  plane  parallel  to  two 
given  lines. 

Otherwise  :  the  equation  of  a  j^lane  through  the  extremity'  of 
y  parallel  to  two  given  lines,  whose  directions  are  given  hy 
a  and  yS,  is  evidently  p  =  y  -\-xa  + 1//3. 

4.  To  find  the  distance  between  two  points. 
a  and  /3  being  vectors  to  the  points, 

y  =  /3-a. 

Squaring 

c^  =  6-  +  a-  —  2  ab  cos  c. 


156  QUATERNIONS. 

5.   A  iilane  being  given  by  tivo  of  its  li7ies,  /3  and  y,  to  icrite 
the  equation  of  a  rigid  line  through  x  perpendicular  to  the  plane. 
Let  OA  =  a.     Draw  two  lines  through  a  parallel  to  /8  and  y. 
Then 

p  =  a  +  x-V/3y (190). 

If  the  plane  is  given  b}-  the  equation  S8p  =  c,  then 

p  =  a  +  a;S (197). 

G.  Find  the  length  of  the  perpendicular  from  a  to  the  j^lane, 
in  the  preceding  example. 

Operating  on  Equation  (197)  with  S  .  8  x 

SSp  =  SSa  +  x5"  =  c, 
or 

xh^  =  c-  S8a ; 
.-.  3;8  =S-^(c-S8a) (198). 

7.  S8(p  — y3)  =  0,  Equation  (189),  being  the  equation  of  a 
plane  through  b  perpendicular  to  8,  to  find  the  distance  from  a 
jjo'int  c  to  the  plane. 

Let  y  =  oc.  The  perpendicular  on  the  plane  from  c,  being 
parallel  to  8,  will  have  for  its  equation 

p  =  y  +  xS. 

To  find  X,  operate  with  S  .  8  x  ,  whence 

S8p  =  SSy  +  xh\ 

or.  from  the  equation  of  the  plane, 

S8y  +  a;82  =  SS/8; 
.-.  x8  =  -S-'HHy-(3), 
and 

xT8  =  TS-^SS(y-^)  =  S[U8  .  (y-^)]. 

8.  Write  the  equation  of  a  plane  through  the  parallels 

p  =  a  +  xp, 
p  =  a'+  xfi. 


APPLICATIONS   TO   LOCI.  157 

9.  Write  the  equation  of  a  plane  tJirough  the  line 

p  =  a  +  Cf/S 

perpendicular  to  the  plane 

SSp  =  0. 

10.  Given  the  direction  of  a  vector-perpendicular  to  a  plane, 
to  find  its  length  so  that  the  plane  may  meet  three  given  planes  in 
a  point. 

Let  8  be  the  given  vector-perpendicular,  and 

Sap  =  a,      S^p  =  &,      Syp  =  c 

the  equations  of  the  given  planes.  If  the  equation  of  the  plane 
be  written 

SSp  =  X, 

then  X  must  haA'e  such  a  value  that  one  value  of  p  shall  satisfy 
the  equations  of  all  four  of  the  planes.  From  Equation  (118) 
we  have 

pSa^y  =  Va^Syp  +  Y^ySap  +  VyaS^p 
=  cYa/8  +  a\liy  +  h\ya. 

Operating  with  S  .  S  x  ,  to  introduce  x, 

icSa/3y  =  cSSa/3  +  aSS/3y  +  &SSya. 

11.  To  find  the  shortest  distance  between  tiuo  given  right  lines. 
Let  the  lines  be  given  by  the  equations 

p  =  a  +  o:/3,  (a) 

p  =  a'  +  x'/3[  (b) 

The  equation  of  a  plane  through  either  line,  as  (6),  parallel  to 
the  other  (a),  is  [Equation  (195)] 

S^;8'(p-a')  =  0.  (c) 

Y(3(3'  is  a  vector-perpendicular  to  this  plane.  Hence,  if  ?/V/8/3' 
be  the  shortest  vector  distance  between  the  lines,  we  have,  since 
a  —  a'—  yY/3(i'  is  a  vector  complanar  with  (3  and  (3', 

S(3ft'ia-a'-y\/3f3')  =  0, 


158  QUATERNIONS. 

or 

S(S/Si8'  +  V^/3')  (a  -«'  -  yV/?/?')  =  0, 
whence 

-2/(V/3/3')^  +  S[V|8;Q'(a-a')]=0; 

or,  dmdingby  T(y^/3'), 

T(t/Y/?;8')=TS[(a-a')U(T/S^')].     .     .(199), 

the  sj'mbol  T  denoting  thtit  onl}-  the  positive  numerical  value  of 
the  sciiUir  is  taken. 

Otherwise :  siuce  the  distance  is  to  be  a  minimum, 

whence 

or 

S(p'-p)/?  =  0     and     S(p'-p)/3'  =  0, 

or  the  shortest  distance  is  their  common  perpendicular,  whose 
length  may  be  found  as  above. 

12.   Give7i  SSi/3  =  (?!  and  SSop  =  do,  the  equations  of  two  planes, 
to  find  the  equation  of  their  line  of  intersection. 
This  equation  will  be  of  the  form 

p  z=  7u8i  +  ?i§2  +  .tVSiSj.  (a) 

To  find  m  and  ??,  we  have,  from  (a), 

S8,p  =  ??i8i- +  ?;SSi82, 
SSjp  =  nh.£  +  ?uS8iS2, 

from  which  we  obtain 

SS,p-?(S8,82      SS2p-«82^ 


m  = 


71  = 


But 


K  S8182 

S8i8oS8]p  —  8i'S82p 


(SS182)-- 8,^82^  =  (V8,8o)^ 
(?,  88, 80-^2  81- 
•*•  "-       (¥8182)^' 


APPLICATIONS   TO   LOCI.  159 

And  similarly 

m=    " 


Substituting  these  values  in  (a) 

which  is  the  equation  of  the  required  line,  a  less  useful  form  than 
those  of  the  two  simple  conditions  of  Art.  70. 

If  the  two  planes  pass  through  the  origin,  then  also  does  their 
hue  of  intersection  ;  and  since  ever}'  line  in  one  plane  is  perpen- 
dicular to  81,  and  ever}'  line  in  the  other  to  So,  VSiSo  is  a  line 
along  the  intersection,  as  in  (o),  and  the  equation  becomes 

P  =  .tV8i82 (200). 

13.    The  planes  being  given  as  in  Equation  (189), 

S8(p-/S)  =  0,  (a) 

SS'(p-iS')  =  0,  (&) 

to  Jin d  the  line  of  intersection. 

The  vector  p  to  an}-  point  of  the  line  must  satisfy  both  («) 
and  (h).  This  vector  may  be  decomposed  into  three  vectors 
parallel  to  8,  8'  and  T88'  which  are  given,  and  not  complanar, 
by  Equation  (118) ;  whence 

pS  .  8S'V88'  =  Sp8T(S' .  T88')  +  Sp8'V(V88' .  8)  +  S(pT88') VS8', 

or,  from  (a)  and  "(6), 

-  p(TV8S')2  =  SSy8T(S' .  VS8')  +  S8';8'V(V8S' .  8)  +  S88'pT88', 

or,  since  835 'p  is  the  only  indeterminate  scalar,  putting  it  equal 
to  X,  we  have 

-  p(TV88')2  =  SS/SV(8' .  V88')  +  SS'^'V(V88' .  8)  +  a;T88: 

If  the  planes  pass  through  the  origin,  in  which  case  (S  and  /?' 
are  zero,  we  have,  as  before, 

p  =z  x\88'. 


160  QUATERNIONS. 

14.  To  tcrite  the  equation  of  a  X)lane  through  the  origin  and 
the  line  of  intersection  of 

S8(p-;8)  =  0,  (a) 

S8'(p-)8')  =  0.  (6) 

If  p  is  such  that  SSp  =  S8/?,  and  also  SS'p  =  SS'^',  then  both  the 
above  equations  will  be  satisfied.     Hence,  from  (a)  and  {h) 

S8pSS'/3'-SS;8SS'p  =  0, 

which  is  also  a  plane  through  the  origin.     This  equation  ma}' 
also  be  written 

S[(8S8'/3'-8'SS/3)p]  =  0, 

which  shows  that 

8S8'/S'-8'SS^ 

is  a  vector-perpendicular  to  the  plane,  and  therefore  to  the  line 
of  intersection  of  (a)  and  (&) . 

15.  To  find  the  equation  of  condition  that  four  j^oints  lie  in 
a  plane. 

If  the  vectors  to  the  four  points  be  a,  /8,  y,  8,  then,  to  meet 
the  condition, 

8  —  a,      8  —  /3,      8  —  y 

must  be  complanar,  and  therefore 

S(S-a)(8-/3)(8-y)  =  0, 
whence 

S8;8y  +  SaSy -f  Sa^8  =  SaySy        .      .       .      (201), 

which  is  the  equation  of  condition. 

Or,  X  and  y  being  indeterminate,  we  have  also 

or 

8  +  (a;-l)a  +  (z/-'^')/?-Z/7  =  0, 
and 

l+(x-l)  +  (y-a;)-y  =  0. 


APPLICATIONS    TO   LOCI.  '  161 

Or,  in  general, 


(202), 


aa  +  bj3-{-cy-{-(l8  =  0] 
a-\-b  +  c-j-d  =  0) 

are  the  sufficient  conditions  of  eomplanarit}'. 

These  conditions  are  analogous  to  Equation  (9). 

16.  Given  the  three  planes  of  a  triedral,  to  find  the  equations 
of  planes  through  the  edges  perpendicidar  to  the  opposite  faces ^ 
and  to  shoiv  that  they  intersect  in  a  right  line. 

Taking  the  vertex  as  the  initial  point,  let 

Sap  =  0,  (a) 

S^p  =  0,  (6) 

Syp  =  0  (c) 

be  the  equations  of  the  plane  faces.  Then  Ta/?  is  a  vector  par- 
allel to  the  intersection  of  (a)  and  (6),  and  V  .  yVa/3  is  a  vector 
perpendicular  to  the  required  plane  through  their  common  edge. 
Hence  the  equation  of  this  plane  is 

S(pY.7Va^)  =  0.  (a') 

Similarl}',  or  by  a  cyclic  change  of  vectors, 

S(pV.aV/3y)  =  0,  (6') 

S(pV./8Vya)  =  0  (c') 

are  the  equations  of  the  other  two  planes. 

If  from  their  common  point  of  intersection  normals  are  drawn 
to  the  planes,  then  are  V  .  yYa^S,  V  .  aY/?y  and  V  .  ^Vya  vector 
lines  parallel  to  them  ;  but,  Equation  (123), 

V(yVa/3  +  aV/3y  +  ySVya)  =  0. 

Hence  these  vectors  are  complanar,  and  the  planes  therefore 
intersect  in  a  right  line. 

Otherwise:  from  Equation  (111) 

V(aY/?y)  =  ySa)8-/3Say; 


162  QUATERNIONS, 

hence,  from  (h'), 

S{pyiaft  -  pfiSay)  =  Sa/SHpy  -  SaySp^  =  0. 

Similarly,  or  b\-  cyclic  permutation, 

S/SySpa  -  HfSaSpy  =  0, 
SyaSp/?  -  Sy/3iipa  =  0. 

But  the  sum  of  these  three  equations  is  identical!}'  zero,  either 
two  giving  the  third  b}'  subtraction  or  addition. 

17.  To  find  the  locus  of  a  point  ichich  divides  all  right  lines 
terminating  in  tioo  given  lines  into  segments  ichich  have  a  com- 
mon ratio. 

Let  DA  and  d'b  (Fig.  C7)  be  the  two 
given  lines,  a  and  yS  unit  vectors  parallel 
to  them,  BA  any  line  terminating  in  the 
given  lines,  and  r  a  point  such  that 
RA  =  ?«Bu.  Assume  dd',  a  perpendicular 
to  both  the  given  lines,  o,  its  middle 
point,  as  the  origin,  and  od  =  8,  od'  =  —  8,  ou  =  p. 
Then 

OA  =  p  +  UA  =  8  +  Xa. 

ou  =  p  +  KB  =  —  8  +  y(i. 
Adding 

2  p  +  UA  +  iiB  =  .Ta  +  y^.  (a) 

But 

RA  -j-  RB  =  RA  =  (  —  p  -f  6  4-  Xa)  , 

m  m 

which  substituted  in  (a)  gives 

p  —  h-xa  =  m{yp  -  p  —  ^)y  (6) 

whence,  since  SS/3  =  SSa  =  0, 

S8p(m  +  l)  =  8-(l-m)  =  c, 

4 

or  the  locus  is  a  plane  perpendicular  to  dd'. 


APPLICATIONS   TO   LOCI.  163 

If  the  given  ratio  is  unit}',  or  br  =  ra,  then  m  =  1  and 

S8/D  =  0,  ' 

and  the  locus  is  a  plane  through  o  perpendicular  to  dd'. 
If  a  and  /S  are  parallel,  then  (b)  becomes 

p  —  8  =  m  {x'a  —  /3  —  8) , 
whence 

SSp{m-\-\)  =  {l-m)S', 

a  right  line  perpendicular  to  dd'.     If  at  the  same  time  m  =  l, 

SS/D  =  0     and     p  =  x''a, 
a  right  line  through  the  origin  parallel  to  the  given  lines. 

IS.  If  the  sxi7ns  of  the  perpendicidars  fro7)i  tivo  given  points  on 
two  given  planes  are  equals  the  sum  of  the  perpendiculars  from 
any  p>oint  of  the  line  joining  them  is  the  same. 

Let  A  and  b  be  the  given  points,  oa  =  a,  ob  =  /?,  and  SS/3=(Z, 
SS'p  =  d'  be  the  equations  of  the  planes ;  8  and  8'  being  unit 
vectors,  so  that  a;8  and  ?/8'  are  the  vector-perpendiculars  from  a 
on  the  planes.     Then 

X  =  Sa8  —  cZ, 
y  =  Sa8'-  d\ 
and 

a;  +  ?/=Sa(8  +  8')-((Z  +  cr). 
Similarly 

.T'+y=S^(8+8')-(cZ  +  (0. 
But,  by  condition, 

Sa(8  + 8')  =  8^(8  +  8'), 
or 

S(/3-a)(8  +  S')=0.  (a) 

The  vector  from  o  to  any  other  point  of  the  line  ab  is 
a  +  2  (/3  —  a)  ;  whence,  for  this  point, 

cc"+y'  =  S[a  +  z(/3-a)](8+8')-(rf  +  fO, 

for  which  point,  smce  (a)  remains  true,  the  sum  therefore  is 
unchan2;ed. 


1G4  QUATERNIONS. 

19.  To  find  the  locus  of  the  middle  poi7its  of  the  elements  of  an 
hyperbolic  paraboloid. 

Let  the  equations  of  the  plane  director  and  rectilinear  direc- 
trices be 

SSp  =  0, 
p  =  a  +  x[i     and     p  =  a'  +  .t'/8I 

Also,  let  CM  =  yiA  be  the  vector  to  the  middle  point  of  an  ele- 
ment so  chosen  that  the  vectors  to  the  extremities  are  a  +  x/3 
and  a'  +  x'[i'.     Then,  since  m  is  the  middle  point, 

2/x  =  a  +  Xy8  +  a'+x'^'.  (a) 

The  vector  element  is 

-  x'fS'  -a'  +  a-\-xp, 

and,  being  parallel  to  the  plane  director, 

SS(-a'  +  aH-.r/?-.r'/3')  =  0. 

This  is  a  scalar  equation  between  known  quantities  from  which 
we  ma}'  find  x'  in  terms  of  x ;  substituting  this  value  in  (a) ,  we 
have  an  equation  of  the  form 

^  =  ai  +  .T/3i, 

or  the  locus  is  a  right  line. 

20.  If  from  any  three  points  on  the  line  of  intersection  of  tivo 
planes,  lines  be  draivn,  one  in  each  plane,  the  triangles  formed 
by  their  intersections  are  sections  of  the  same  pyramid. 

The  Circle  and  Sphere. 

78.   Er/uations  of  the  circle. 

The  equation  of  the  circle  may  be  written  under  various 
forms.  If  a  and  /3  are  vector-radii  at  right  angles  to  each  other, 
and  Ta  =  T/?,  we  ma}-  write 

p  =  cos^  .  a  +  sin^  .  ;8    ....     (203), 

in  terms  of  a  single  variable  scalar  6. 


APPLICATIONS    TO    LOCI. 


165 


If  a  and  (3  are  unit  vectors  along  the  radii, 
or,  since  x^  -(- 1/-=  ?•-, 

Tlie  initial  point  being  at  the  center, 


T^=l 


p-  =  -r 


(204), 


(205) 


are  evidentl}'  all  equations  of  the  circle. 

If  o  (Fig.  68)  be  any  initial  point,  c  the  center,  to  which  the 
vector  oc  =  y,  p  the  variable  vector  to 
any  point  p,  cp  =  a,  then 


whence 


p  —  y=a, 

(p-y)^  =  -7-'.      .(206), 

the  vector  equation  of  the  circle  whose 
radius  is  r. 

If  Ty  =  c,  it  ma}'  be  put  under  the  form     q 

/D2-2Spy  =  c--?'2 (207). 

If  the  initial  point  is  on  the  circumference,   we  still  have 
(p  —  y)-  =  —  ?"  ;  but  -f  =  —  7~,  hence 

p--2Spy  =  0 (208), 

or,  since  in  this  case  Spy  =  Spa, 

p2_2Spa  =  0 (209). 

79.  Equations  of  the  sphere. 

This  surface  may  be  conveniently  treated   of  in  connection 
with  the  circle  ;  for,  since  nothing  in  the  previous  article  restricts 
the  lines  to  one«plane,  the  equations  there  deduced  for  the  circle, 
are  also  applicable  to  the  sphere. 


166 


QUATERNIONS. 


Another   convenient   form   of  the   equation   of  a   sphere   is 
(Fig.  68) 

T(p-y)  =  Ta (210), 

the  center  being  at  the  extremit}-  of  y  and  Tu  the  radius. 

80.    Tangent  line  and  2^1  cine. 

A  vector  along  the  tangent  being  dp,  we  have,  from  Equation 
(203), 

dp  =  —  sin^  .  a  +  cos^  .  /?, 

and  for  the  tangent  Hne  Tr  —  p  +  xdp, 

7r=cos^  .  a  +  sin^  .  /?  +  .r[-sin^  .  a  +  cos^  .  ^]    (211), 

where  tt  is  any  vector  to  the  tangent  hne  at  tlie  point  corre- 
sponding to  6. 

From  the  above  we  have  directl}- 

Spdp  =  0,  v^ 

or  the  tangent  is  perpendicular  to  the  radius  vector  drawn  to  the 
point  of  tangency. 

By  means  of  this  property  we  ma}' 
rB  write  the  equation  of  the  tangent  as 
follows  :  let  tt  be  the  vector  to  any  point 
of  the  tangent,  as  b  (Fig.  69),  c  being 
the  initial  point  and  p  the  vector  to  P, 
the  point  of  tangenc}'.     Then 

Sp(,r-p)  =  0, 

S/JTT  =   —  >- 


(212), 


S-  =  l 
P 


are  the  equations  of  a  tangent.  Since  nothing  restricts  the  line 
to  one  plane,  they  are  also  the  equations  of  the  ttingent  plane  to 
a  sphere. 


APPLICATIONS   TO   LOCI.  167 

The  above  well-known  propert}'  may  also  be  obtained  by 
differentiating  Tp  =  Ta  ;  whence,  Art.  67,  2, 

Spdp  =  0, 

and  therefore  p  is  perpendicular  to  the  tangent  line  or  plane. 

81.  Chords  of  contact. 

In  Fig.  69  let  cb  =  /3  be  the  vector  to  a  given  point.  The 
equation  of  the  tangent  bp  must  be  satisfied  for  this  point ; 
hence,  from  Equation  212, 

S/Sp  =  -  7-2, 
or 

S/3o-  =  -?-2 (213), 

which  is  equally-  true  of  the  other  point  of  tangency  p,'  and  being 
the  equation  of  a  right  line,  it  is  that  of  the  chord  of  contact  ppI 
And  for  the  reason  previously  given,  it  is  also  the  equation  of 
the  plane  of  the  cu'cle  of  contact  of  the  tangent  cone  to  the 
sphere,  the  vertex  of  the  cone  being  at  b. 

82,  Exercises  and  Problems  on  the  Circle  and  the 
Sphere. 

In  the  following  problems  the  various  equations  of  the  plane, 
line,  circle  and  sphere  are  emploj-ed  to  familiarize  the  student 
with  their  use.  Other  equations  than  those  selected  in  any 
special  problem  might  have  been  used,  leading  sometimes  more 
directly  to  the  desired  result.  It  will  be  found  a  useful  exercise 
to  assume  forms  other  than  those  chosen,  as  also  to  transform 
the  equations  themselves  and  interpret  the  results.  Thus,  for 
example,  the  equation  of  the  circle  (209), 

p^-2^pa  =  0 

may  be  transformed  into 

Sp(p-2a)  =  0, 


168 


QUATEKNIONS. 


which  gives  immediately  (Fig-  7U)  the  property-  of  the  circle, 
that  the  angle  inscribed  in  a  semi-circle  is 
a  right  angle.  Obviousl}',  this  includes  the 
case  of  chords  drawn  from  an}-  point  in  a 
sphere  to  the  extremities  of  a  diameter,  and 
the  above  equation  is  a  statement  of  the  prop- 
osition that,  p  being  a  variable  vector,  the 
locus  of  the  vertex  of  a  right  angle,  whose 
sides  pass  through  the  extremities  of  a  and 
^  —  a,  is  a  sphere. 

Again,  with  the  origin  at  the  center,  we  have  (Fig.  71), 

(p  +  a)  +  (a-p)=2a, 

and,  operating  with  x  S  .  (p  —  a) , 
S(p  +  a)(p-a)  =  0; 


.'.  p  is  a  right  angle.     This  also  follows  from 
Tp=Ttt,  whence  p-=a-  and  S(p+a)(p— a)  =  0. 
Again,  from  Tp  =  Ta, 

T(p  +  a)(p-a)=2TTap. 

The  first  member  is  the  rectangle  of  the  chords  pd,  pd'  (Fig.  71), 
and  the  second  member  is 


2  CD  ,  OP  sinDOP. 

Hence  the  rectangle  on  the  chords  drawn  from  an}-  point  of  a 
circle  to  the  extremities  of  a  diameter  is  four  times  the  area  of 
a  triangle  whose  sides  are  p  and  a. 

Also,  from  Tp  =  Ta, 


and  for  au}-  other  point 


P^  =  -A 


•••  S(p'+p)(p'-p)  =  0. 

But  p'—  p  is  a  vector  along  the  secant,  and  p'-{-  p  is  a  vector 
along  the  angle-bisector ;  uow  when  the  secant  becomes  a  tan- 


APPLICATIONS    TO    LOCI. 


169 


gent,  the  angle-bisector  becomes  the  radius  ;  therefore  the  radius 
to  the  point  of  contact  is  perpendicular  to  the  tangent. 

1 .    The  angle  at  the  center  of  a  circle  is  double  that  at  the  cir- 
cumference standing  on  the  same  cax. 


Tp  =  Ta, 


We  have 

and  therefore,  Art.  56,  18, 

p  =  (p  +  a)-ia(/3+a), 
whence  the  proposition. 

2.    In  any  circle,  the  square  of  the  tangent  equals  the  product 
of  the  secant  and  its  external  segment. 

Fig.  69  (bis). 

In  Fig.  69  we  have  p 

CB  =CP  +PB, 
.'.    CB-=CP-4-PB^, 


or 


PB-=CB-'— CP^ 

=  CB^— CD^,  as  lines, 
=  BD  .  bd'. 


3.    Tlie  right  line  joining  the  points  of  intersection  of  two  circles 
is  perpendicular  to  the  line  joining  their  centers. 

Let  (Fig.  72)  cc'  =  a,  cp  =  p,  cp'  =  p',  and  r,  r'  be  the  radii 
of  the  circles.     Then 


also 

Hence 
or 


(p_a)-  =  -r'-; 


{p'-ay  =  -r'^. 
Spa  =  Sp  a, 

Sa(p-p')  =  0; 
hence  pp'  and  cc'  are  at  right  angles. 


170  QUATERNIONS. 

4.  A  chord  is  drawn  paruUel  to  the  diameter  of  a  circle;  the 
radii  to  the  extremities  of  the  chord  make  equal  angles  with  the 
diameter. 

If  p  and  p'  be  the  vector-radii,  2  a  the  vector-diameter,  then 
xa  =  the  vector-chord,  and 

(p'-Xa)-  =  -'t^, 

(p-j-a;a)-  =  -)-, 

whence  the  proposition. 

5.  If  ABC  is  a  triangle  inscribed  in  a  circle,  shoiv  that  the  vector 
of  the  product  of  the  three  sides  in  order  is  parallel  to  the  tangent 
at  the  initial  p>oint.     [Compare  Ai't.  55.] 

If  AB  =  /3,  CA  =  y,  and  o  is  the  center  of  the  cii'cle,  then 

-T(AB  .  BC  .  Ca)  =  V  .  /3(^  +  y)y 

c  and  B  being  points  of  tlie  circumference  satisfying 
p-  — 2Spa=0  [Eq.  (209)],  substituting  and  operating  with 
S  .  aX 

S  .  aV  (ab  .  BC  .  Ca)  =  2  Sa/3Say  -  2  Sa^Say  =  0. 

Hence  y(AB  .  bo  .  ca)  is  perpendicular  to  a,  or  parallel  to  the 
tangent  at  a. 

6.  The  sum  of  the  squares  of  the  lines  from  any  x)oint  on  a 
diameter  of  a  circle  to  the  extremities  of  a  2^arallel  chord  is  equal 
to  the  sum  of  the  squares  of  the  segments  of  the  diameter. 

Let  pp'  (Fig.  73)  be  the  chord  parallel  to  the  diameter  dd' 
Fiji.  73.  o  the  given  point,  and  c  the  center  of  the 

circle.     Let  cp  =  p,  cp'  =  p,  oc  =  a,  op  =  ft 
and  op'  =  /?!     Then 

^^'  op2  =  -^^  =  -(a=  +  2Sap+p2), 

.Ov'2  =  -/3'-=-{a'+2fiap'+p")  ; 
.  • .  OP-  +  OP'-  =  2  OC^  +  2  DC-  -  2  (Sap  -I- Sap'). 


APPLICATIONS   TO   LOCI.  171 

But 

S(p-p')(p  +  p')  =  S(p+p>-a=0. 

Therefore 

Sap  +  Sap'  =  0, 

and 

op^  +  op'-  =  DO-  +  od'^. 

7.  To  find  the  intersection  of  a  plane  and  a  sphere. 

Let  p-  =  —  ^-^  be  the  equation  of  the  sphere,  8  a  vector-perpen- 
dicular from  its  center  on  the  plane  and  TS  =  d.  Then,  if  /3  be 
a  vector  of  the  plane, 

P  =  ^  +  IB. 

Substituting  in  the  equation  of  the  sphere,  since  S(38=  0,  Tve 
have 

the  equation  of  a  circle  whose  radius  is  V?*^  —  rf^,  and  which  is 
real  so  long  as  d  <  r. 

8,  To  find  the  intersection  of  two  spheres. 

Let  the  equations  of  the  given  spheres  be  (Eq.  207) 

p2  -  2  Spy  =  c-  -  7-2, 
p--2Spy'=c'--r'% 

Subtracting,  we  have 

2  Sp(y  —  y')  =  a  constant. 

The  intersection  is  therefore  a  circle  whose  plane  is  perpen- 
dicular to  y—  y'  the  vector-line  joining  the  centers  of  the  spheres. 
Assuming  (Eq.  210) 

T(p-y)  =  Ta     and     T(p-y')  =  Ta,' 
show  that  2  Sp  (y  —  y ')  =  a  constant,  as  above. 


172  QUATERNIONS. 

9.  Tlie  planes  of  intersection  of  three  spheres  intersect  i?i  a 
right  line. 

Lot  y',  y".  y'"  be  the  vector-lines  to  the  centers  of  the  spheres, 
and  their  equations 

p--2Spy'=c', 
p'-2Spy"  =  c", 
p^-2Spy"'=c"'. 

The  equations  of  the  planes  of  intersection  are,  from  the  pre- 
ceding problem, 

2Sp{y'-y")  =c"-c\  (a) 

2Sp{y'-y"')=c"'-c',  (b) 

2Sp(y"-y"')=c"'-c".  (C) 

Now,  if  p  be  so  taken  as  to  satisfy  (n)  and  (&),  we  shall 
obtain  their  line  of  intersection.  But  if  p  satisfies  (a)  and  (6), 
it  will  also  satisfy  their  difference,  which  is  (c) ;  the  planes  there- 
fore intersect  in  a  right  line. 

10.  To  find  the  locus  of  the  intersections  of  perpendiculars  from 
aJixedx>oint  ujwn  lines  through  another  fixed  point. 

Let  p  and  v'  be  the  points,  rp'  =  a,  and  8  a  vector-perpen- 
dicular on  any  line  thi'ough  pj  as  p  =  a  -f-  xft.     Then 

^  =  a  +  7//5, 

and  operating  with  S  .  8  x 

h-  =  SSa, 

which  is  the  equation  of  a  circle  (Eq.  209)  whose  diameter  is  rv'. 

11.  From    a  fixed  point  p,   lines   are   draicn  to  j^oi^its,  as 

p',  p",  of  a  given  right  line.     Required  the  locus  of  a  point  o 

on  these  lines,  such  that  pp'  .  po  =  ?u^. 

Let  the  variable  vector  PO=p  ;  then  rv'=xp.    By  the  condition 

T(pp' .  po)=     ?u^, 
or 

T{xp  .  p)=     m^; 


APPLICATIONS   TO   LOCI.  173 

If  8  be  the  vector-perpendicular  from  p  on  the  given  line, 

and  T8  =  d, 

SS(a;p-8)  =  0, 
or 

ajSSp  =  —  c?- ; 

..P=-S8p, 

hence  the  locus  is  a  circle  through  p. 

12.  If  through  any  2'>oint  cJiords  be  draion  to  a  circle^  to  find 
the  locus  of  the  intersection  of  the  pairs  of  tangents  draion  at  the 
2)oints  of  section  of  the  chords  and  circle. 

Let  the  point  a  be  given  b}'  the  vector  oa  =  a,  o  being  the 
initial  point  taken  at  the  center  of  the  circle.  Let  p'  =  or  be 
the  vector  to  one  point  of  intersection  r.  The  locus  of  r  is 
required.     The  equation  of  the  chord  of  contact  is  (Eq.  213) 

Sp'o-  =  —  7", 

which,  since  the  chord  passes  through  a,  may  be  written 
Sp'a  =  —  r-, 

where  a  is  a  constant  vector.     The  locus  is  therefore  a  straight 
line  perpendicular  to  oa  (Eq.  180). 

13.  To  find  the  locus  of  the  feet  ofperjoendicidars  drcavn  through 
a  given  point  to  planes  passing  through  another  given  point. 

Let  the  initial  point  be  taken  at  the  origin  of  perpendiculars, 
a  the  vector  to  the  point  through  which  the  planes  are  passed, 
and  8  a  perpendicular.     Then 

SS(8-a)=0, 

or 

82_Sa8=0 

is  true  for  any  perpendicular.    Hence  the  locus  is  a  sphere  whose 
diameter  is  the  line  joining  the  given  points. 


174  QUATERNIONS. 

Otherwise  :  if  the  origin  be  taken  at  the  point  common  to  the 
planes,  and  the  equation  of  one  of  the  planes  is  SSp  =  0,  then 
the  vector-perpendicular  is  (Eq.  198) 

S-'SSa, 

and,  if  p  be  the  vector  to  its  foot, 

p  =  a  —  8~^  S8a, 
or 

p  —  a  =     —  8~'  S8a, 
■whence 

(p-ay=    8-\s8ay, 

and 

Sap  —  a"  =  —  8~"(S8a)'. 

Adding  the  last  two  equations 

p'  —  Sap  =  0, 
or 

T(p--U)  =  T^a, 

which  is  the  equation  of  a  sphere  whose  radius  is  T-  and  center 

CI 

is  at  the  extremity  of  -,  or  whose  diameter  is  the  line  joining 

the  points. 

14.  To  find  the  locus  of  a  point  p  wliich  divides  any  line  os 
draicn  from  a  given  point  to  a  given  plane,  so  that 

OP  .  OS  =  m,  a  constant. 

Let  OP  =  p  and  os  =  o- ;  also  let  SSo-  =  c  be  the  equation  of  the 
plane.    We  have,  by  condition, 

TpTa-  =  m, 

and 

Up  =  Ucr ; 

.-.   Tcr  =  ^, 
Tp 
and 

??iUp 

"^ 

mp 


APPLICATIONS   TO   LOCI.  175 

Substituting  in  the  equation  of  the  plane 

mSSp  +  cp^  =  0, 

which  is  the  equation  of  a  sphere  passing  through  o  and  having 
—  for  a  diameter, 

OD 

15.  To  find  (lie  locus  of  a  point  the  ratio  ofwJiose  distances 
from  tivo  given  points  is  constant. 

Let  o  and  a  be  the  two  given  points,  oa  =  a,  or  =  p,  r  being 
a  point  of  the  locus.  Then,  by  condition,  if  m  be  the  given 
ratio, 

T(p  — a)  =  mTp,       __ 
whence 

p^—  2  Spa  +  a-  =  ni^p^i 

(1  —  m")p^  =  2  Sap  —  a? 

=  2Sap-J^a^ 

1  —  nr 
or 

2  _     2  Sap    _| a?  _        m^o?       , 

1  —  m-      (1  —  ni')-      (1  —  m-)- 


1— my         1— m^ 

which  is  the  equation  of  a  sphere  whose  radius  is  T^; 5  a,  and 

^        1  — m" 

whose  center  c  is  on  the  line  oa,  so  that  oc  = ;  a.   (Eq.  210) . 

1—m- 

16.    Given  two  j^oints  a  and  b,  to  find  the  locus  ofv  lohen 

AP^  +  BP-  =  0P^ 

o  being  the  origin,  let  oa  =  a,   ob  =  ^,  op=p.     Then,  by 

condition, 

p2  =  (p-a)2  +  (p-/3)S 
whence 

p2_2Sp(a  +  /3)  =  -(a--f^=^), 

[p-(a  +  ^)]-=2Sa;8, 
T[p-(a  +  ^)]  =  V-2Sa^, 


176  QUATERNIONS. 

which  is  the  equation  of  a  sphere  whose  center  is  at  the  extremity 
of  (a  +  )8),  if  Sa/3  is  negative,  or  the  angle  aob  acute.  If  this 
angle  is  obtuse,  thei'e  is  no  point  satisfying  the  condition.  If 
AOB  =  90°,  the  locus  is  a  point. 

83.   Exercises   in  the  transformation   and   interpretation   of 
elementary  symbolic  forms. 

1 .  From  the  equation 

derive  in  succession  the  equations 

T(p  +  a)  =  T(p-a)     and     T^-^=  1, 

p  —  a 

and  state  what  locus  they  represent. 

2.  From  the  equation 

K-+-=0 

a       a 

derive  sj'mbolicalh'  the  equations 
ap  +  pa  =  0,   S-  =  0,  SU-  =  0,  (U^)=-l,  and  TVU-  =  1, 

and  interpret  them  as  the  equations  of  the  same  locus. 

3 .  Transform 


to  the  forms 
and  interpret. 


p  —  a 
S =  0 


P  pa 

S-=l     and     SU-=T-. 


4.    Transform  S^ — ^  =  0  to  S-=S-'  'ind  interjoret. 


a  a 


5.  Transform   (p -/5)- =  (p  -  a)^   to    T  (p -^)  =  T  (p -a), 
and  interpret. 

6.  "What  locus  is  represented  by  K =  0  ? 


APPLICATIONS    TO    LOCI.  177 


7.  What  by  Q'=  -  1  ?  By  Q'=  -  a^ ? 

8.  Whatby  U^=U^?  Up  =  U/3?     U^=l? 

P  /^. 

9.  U-  =  -U-? 


10.   (U-)  =u-? 


11.    V ^  =  0?     V-  =  V-? 


12.    V-=0? 


13.   -K-  =  fr? 
a      a 


14.  su-  =  su-?    su-  =  -su-?    (su-)=(su-)? 


15.  Tp  =  l? 

16.  Transform  (p  —  a)-  =  a-  to  T(p  —  a)  =  Ta,  and  interpret. 

17.  Under  what  other  form  may  we  write  (p  —  ay=(j3—  aY? 
Of  what  locus  is  it  the  equation  ? 

18.  p^  +  a-  =  0?  p2  +  1  =  0 ?  Translate  the  latter  into  Car- 
tesian coordinates,  b}'  means  of  the  trinomial  form,  and  so  detei'- 
mine  the  locus  anew. 

19.  T(p-^)  =  T(/3-a)? 

20.  Compare  SU  -  =  T-  and  S-  =  1  with  the  forms  of  Ex.  3. 

pa  p 

21 .  What  locus  is  represented  by  S/5p  +  p-  =  0  when  T/3  =  1  ? 

22.  (S^Y-f'v^'l  =  l? 


23.    (T-V=-l? 


24.    Sliow  that  V  .  Ya^Yap  =  0  is  the  equation  of  a  plane. 
What  plane?     [Eq.  (112)]. 


178  QUATERNIONS. 

The  Conic  Sections. 
Cartesian  Forms. 
84.    TJte  Parabola. 
Resuming  tlie  general  form  of  the  equation  of  a  plane  curve 

p  =  xa-\-  y(3, 

from  the  relation  y-  =  2p.r,  we  obtain 

P  =  ^^a  +  y(3 (214) 

for  the  vector  equation  of  the  parabola  when  the  vertex  is  the 
initial  point.  If  tlie  latter  is  taken  anywhere  on  the  curve,  from 
the  relation  y-  =  'Ij^'x,  we  obtain 

P  =  ^«  +  2//? (215); 

and  if  the  initial  point  is  at  the  focus,  then  y"^  =  2px  -\- p-  gives 

P  =  ^{y--ir)a  +  y(B      ....    (216); 
or  again,  in  terms  of  a  single  scalar  t, 

P  =  \o.  +  tp (217). 

In  Equations  (214),  (21.5)  and  (21 G),  a  and  /?  are  ?<?»Y  vectors 
parallel  to  a  diameter  and  tangent  at  its  vertex,  being  at  right 
angles  to  each  other  in  Equations  (214)  and  (216)  ;  in  Equation 
(217)  a  and  /?  are  any  given  vectors  parallel  to  a  diameter  and 
tangent  at  its  vertex,  the  initial  point  being  on  the  curve. 

85.    Tangent  to  the  parabola. 

From  Equation  (216)  we  have  for  the  vector  along  the  tangent 
(Art.  62) 


APPLICATIONS   TO   LOCI.  179 

and,  therefore,  the  equation  of  the  tangent  is 

''  =  ^^y'-i''^''  +  ^(^  +  ^^(j,"'  +  (^)     •     •   (218). 
From  Equation  (217)  the  vector  along  the  tangent  is 

ta  +  fS, 

and  the  equation  of  the  tangent  is 

7r  =  ^^a-ht/3  +  x{ta+(3)     ....    (219). 

If  p  be  the  vector  to  a  point  on  the  diameter  of  a  parabola,  the 
point  being  given  by  the  equation 

p  =  ma  +  n/3,  (a) 

and  a  tangent  to  the  curve  be  drawn  tln'ough  this  point,  then 
(a)  must  satisfy'  the  equation  of  the  tangent-line  and 

ma  +  ?iy8  =  -a  +  t/3  -\-  x(ta  -{-  (3)  , 

whence  ,2 

m  =  -  +  xt     and     n  =  t  4-  x, 

or  ,— 

t  =  n  ±y)i' — '2m\  (&) 

hence,  in  general,  two  tangents  can  be  drawn  to  the  curve  through 
the  given  point.  When  n^  =  2m,  they  coincide  ;  in  this  case, 
from  (a),  ^^2 

p^—a-\-nf3, 

the  point  being  on  the  cui've.  If  2m>n^,  t  is  imaginary,  and 
no  tangent  can  be  drawn  ;  in  this  case  (a)  becomes 

P=[^  +aja  +  n^, 
the  point  being  within  the  curve. 


180 


QUATERNIONS. 


86.   Examples  on  the  parabola. 

1 .    The  intercept  of  the  tangent  on  the  diameter  is  equal  to  the 


Fig.  74. 


abscissa  of  the  x>oint  of  contact. 

Since  the  tangent  is  parallel  to 

the  vector  ta-\-  ft,  or  to  an}'  multiple 

of  it,  it  is  parallel  to  t-a  +  tft  or  to 

/-  t- 

-a  +  i/3  +  -a,  that  is,  to  (Fig.  74) 


But 


OP  +  ox. 

TP  =  TO  +  OP  ; 
TO  =  ox. 


2.  If  from  any  point  on  a  di- 
ameter 2)roduced ,  tangents  be  draicn, 
the  chord  of  contact  is  piarallel  to  the 
tangent  at  the  vertex  of  the  diameter. 

If   t'   and   <"  correspond    to  the 
points  of  tangenc}',  we  have  for  the  A-ector-chord  of  contact 


P'-P 
which  is  parallel  to 


^la  +  t'ft-^^a-t"(i. 


P  + 


f'+t" 


or,  from  Equation  (6),  Art.  85,  to 

/3  +  na^ 
which  is  independent  of  m. 

3.    To  find  the  locus  of  the  extremity  of  the  diagoncd  of  a  rect- 
angle whose  sides  are  two  chords  draicn  from  the  vertex. 

Let  OP  and  op'  be  the  chords.     Then 


OP  = 


P  =  lj^a  +  7jft, 


or'=p'='fa-y% 
2p 


(«) 
(&) 


*  APPLICATIONS   TO   LOCI.  181 

The  vector-diagonal  w'  is  p  +  p'  or 

which  may  be  put  uuder  tlie  form  of  the  equation  of  the  parabola 

2  till' 
hy  adding  and  subtractmg  -^  a,  giving 

But,  by  condition,  S/ap' =  0.     Hence,  from  (a)  and  (&),  Sa/3 
being  zero, 

yy'-^^=^^   •••  2/i/'=(2p)^  (^) 

which  in  (c)  gives 

Changing  the  origin  to  the  extremity  of  4j3a, 

.     {y-iir 


2p 


■a  +  {y-y')fi. 


Hence  the  locus  is  a  similar  parabola  whose  vertex  is  at  a 
distance  of  twice  the  parameter  of  the  given  parabola  from  its 
vertex. 

Moreover,  from  (f?),  xx'  =  {2py.  Hence  the  parameter  is  a 
mean  propoHional  between  the  ordinates  and  the  abscissas  of  the 
extremities  of  chords  at  right  angles. 

4.  If  tangents  be  drawn  at  the  vertices  of  an  inscribed  triangle, 
the  sides  of  the  triangle  produced  ivill  intersect  the  tangents  in  three 
points  of  a  right  line. 

Let  opp'  (Fig.  74)  be  the  inscribed  triangle,  and  one  of  the 
vertices,  as  o,  the  initial  point.  Then,  for  the  points  p  and  p' 
respective!}' ,  we  have 

P  =  ^'^  +  '/5' 


182 


QUATERNIONS. 


Let  TTi,  7r2,  TTg  be  the  vectors  to  the  points  of  iutorseetiou  ;  then 


t- 


Also 


TTi  =  OP  +  PSi  =  -  a  +  //?  +  X{ta  +  (3)  . 


■7ri  =  x'or'=x'(— a +  t'f3]; 


t-  r't'- 


^ 


•Zit'-t'' 


Hence 


'      2tt'-t'\2  ^j      2t-t'\2        V 


In  a  similar  manner 


But 


Also 


'==iFb(l°+^ 


TTg  =  OP  +  .?/pp'  =  ov+y(p'—  p) 


=  U  +  //?  +  ^ 


L_±a  + {t'-t)/3 


tt' 


t  +  t' 


Hence 


Now 


-'^-':^,_  2«z:_«^,_  ^V3=/^-Wy8V 


t 
Also 


2t-t'      2t'-t     t-—t'- 


=  0. 


t  t'  W 

Hence  tti,  ttj  and  TTg  terminate  in  a  straight  line. 


APPLICATIONS   TO   LOCI. 


183 


5.    The  jyfincqyal  tangent  is  tangent  to  all  circles  described  on 
the  radii  vectores  as  diameters. 

Fig.  75. 

Let  AP  =  p  (Fig.  75) ,  a  and  /3 
being  unit  vectors  along  the  axis   d 
and  principal  tangent.     Tlien,  if 
tlie  circle  cut  the  tangent  in  x, 
and  TC  be  drawn  to  the  center, 

T(tc)  =  T(fc)  =  T(^FP)  ; 
.'.  TC-  =  i(p  —  ma)". 

Also 

TC  =  TA  +  AF  +  FC 

=  —  ;^/3  +  i?ia  +  i  (p  —  '>na) , 

TC-  =  \^—z(3-\-Vla-j-^{p  —  ma)Y. 

Equating  these  values  of  tc", 
we  have,  since  fi/3a=  0, 

2-/3"  —  z^f3p  +  mfiap  =  0, 


■^y-f- 


0, 


which  o-ives  but  one  value  for  z. 


6.    To  find  the  length  of  the  curve. 

It  has  been  seen  (Art.  62)  that,  if  p  =  (f)(t)  be  the  equation 
of  a  plane  curve,  the  differential  coefficient  is  the  tangent  to  the 
curve.  Hence,  if  this  be  denoted  by  p'=(f>'(t),  Tp'dt  is  an 
element  of  the  curve  whose  length  will  be  found  b}-  integrating 
Tp'  with  reference  to  the  scalar  variable  involved  between  proper 
limits  ;  or 


-..=jrv. 


For  the  parabola 


P=^«+2/^» 
22) 


184 


QUATERNIONS. 


we  have 


7.    To  Jin  (I  the  area  of  the  curve. 

AVith  the  notation  of  the  previous  example,  twice  the  area 
swept  over  by  the  radius  vector  will  be  nieasured  hy  (Art.  41,  7) 
TYpp'dt.  The  area  will  then  he  found  by  integrating  T\pp'  with 
reference  to  the  involved  scalar  between  proper  limits  and  taking 
one-half  the  result ;  or 

A  -  A,  =  I-  C'lYpp'. 

For  the  parabola 

A 

A 

or,  since  ayS  =  90°, 


V(|^«  +  !//i)(?«+/J 


.J 


From  the  origin,  where  y^  =  0,  to  an}'  point  whose  ordinate  is 
y,  the  area  of  the  sector  swept  over  b}'  p  is  -; — y^  =  -^xy ;  adding 

the  area  ^xy  of  the  triangle,  which,  with  the  sector,  makes  up 
the  total  area  of  the  half  curve,  we  have  ^xy,  or  two-thirds  that 
of  the  circumscribing  rectangle.  The  origin  ma}'  be  changed  to 
an}'  point  in  the  plane  of  the  curve,  to  which  the  vector  is  y,  by 
substituting  the  value  p  =  y  +  p^  in  the  equation  of  the  curve, 
Pi  being  the  new  radius  vector  ;  we  ma}'  thus  find  any  sector  area 
limited  by  two  positions  of  pi,  the  vertex  of  the  sector  being  at 
the  new  origin.  Thus,  transferring  to  an  origin  on  the  principal 
tangent,  distant  b  from  the  vertex,  p  =  hp  -\- pi\  which,  in  the 
equation  of  the  parabola,  gives 


Pi  =  ^«  +  (y-&)A 


P^=ja  +  li; 


Fig.  7G. 


APPLICATIONS   TO   LOCI.  185 

integrating,  as  before,  between  the  limits  y  =  h  and  ?/  =  0, 

2n.   Relations  between  three  intersecting  tangents  to  the  Parabo- 
la.    ["Am.  Journal  of 
Math.,"  vol.  i.  p.  379. 
M.     L.     Holman     and 
E.  A.  Engler.] 

Let  pi,  p2i  Ps  he  the 
vectors  to  the  three 
points  of  tangenc}^  Pi, 
P2,  Ps  [Fig.  76],  and 
TTj,  TTg,  TTs  the  vectors  to 
Si,  S2,  S3,  the  points  of 
intersection  of  the  tan 
gents.  Resuming  Equa- 
tion (216),  where  the 
focus  is  the  initial 
point,  and  a  and  (i  are 
unit  vectors  along  the 
axis  and  the  directrix, 


2p 


(«). 


Since  p'  =  —  (Tp)",  and  Sa/3  =  0,  we  have  for  the  three  points 


Pi,  P2,  P3 


Tpi  =  — (z/r+ir) 
22) 

Tp2  =  i-(H+p^) 
•22) 

22) 


(&). 


The  vector  alons;  the  tangent  is 


|«+^, 


186 

and  therefore 


QUATERNIONS. 


^i  =  P2  +  P2S1  =  —Oji  -iy')o.  +  y2^  +  z  ( -a  +  (A, 
7^1  =  P3  +  P3S1  =  —  {yi  -p^)o-  +  Vzl^  +  yy-o.  +  ft] ; 

whence,  equating  the  coefficients  of  a  and  /8, 

2  =  1(2/3-2/2),     ■it^  =  10/2-^3), 

the  cj'clic  permutation  of  the  sulo- 


whence,  substituting,  and  b; 
scripts, 

7^1=  — (2/3  2/2- 

2i) 

T2  =  — (2/12/3- 
7^3  =  7- (2/2  2/1- 

2i9 


From  (6) 


TpiTp,= 


4p' 


f^^Tp3  =  J^ 


and  from  (c) 


(Tm)=  = 

(T.,)^  = 
(T:r3)^  = 


4p2 

_1_ 

4p2 

_1_ 

1 

4ir 


and  from  (d)  and  (e) 


(Ttts 
(Ttt^ 
(Ttt, 


J 


'^')a  +  ^(y2  +  y3)/3 

>')a  +  ^(2/3  +  2/i)/? 
'^')a  +  ^(2/i+2/2)^ 

2/i'+P')(2/2Hir) 

y2'+p')(2/3'+F) 

2//+i>')(2/i^+/) 

2/2'+i>=')(2//+F) 
2/3'+p')(2/i'+i>')    - 

yi+p-){yi+p-) 


(c). 


(f?), 


(e), 


J 


2  =  Tp,Tp/ 

2=T/D,Tp3 
2  =  Tp2T/33. 


(/). 


APPLICATIONS   TO   LOCI. 


187 


From  (c) ,  it  appears  that  the  distance  of  the  x>oint  of  intersec- 
tion of  two  tangents  from  the  axis  is  the  arithmetical  mean  of  the 
ordinates  to  their  points  of  contact.  From  (/) ,  that  the  distance 
from  the  focus  to  the  point  of  intersection  of  tico  tangents  is  a 
mean  proportional  to  the  radii  vectores  to  the  points  of  contact. 

1st.    If  p2  becomes  a  multiple  of  /3, 

P2  =  T-  {yi  -ir)  a.  +  2/2/5  =  2/5 ; 
2p 

.'.  z  =  y.2  =  ±p. 

Or,  the  piarameter  is  the  double  ordinate  tnrough  the  focus,  or 
tivice  the  distance  from  the  focus  to  the  directrix. 


Fig.  77, 


2cl.    If  pi  is  the  multiple  of  pa  (Fig.  77),  then  po  —  /d^  is  a  focal 

chord,  and 

^p2  —  Pll 
or,  from  (a). 


^(2/2' -i^')a+ 2/2/5 
2p) 


=  ^(yi--2r)a  +  y^fi; 
2p 


whence 


yi-p"    2/2' 


188  QUATERNIONS. 

or 

and 

yiy2+2r  =  0.  (rj) 

From  (a)  and  (c) 

s-3pi  =  -  —  (z/22/1  -  p-) — (.'/i-  -  ;>')  -  K^i  +  !/2)yi 

2i>  -22) 

=  -^.(y'  +  p-)  (yiy^  +  jr)  =  0 ;  (h) 

or,  a  line  from  the  focus  to  the  intersection  of  the  tangents  at  the 
extremities  of  a  focal  chord  is  perpendicular  to  the  focal  chord. 
The  vectors  along  the  tangents  are 

pi  —  TTg         and         p.,  —  TTg, 

and,  from  (h), 

S(pi  —  Ta)  (P2  —  ■^n)  =  Sp,  po  +  7^3"  =  0? 

or,  the  tangents  at  the  extremities  of  the  foocd  chord  are  perpen- 
dicular to  each  other. 
Since,  from  (<;), 


we  have 


!/i!/2  =  -iA 

^z  =  —  (yi  ?/2  -  2^')  a  +  H^/i  +  2/2)  y3 

•2p 

or,  the  tangents  at  the  extremities  of  a  foccd  chord  intersect  on 
the  directrix. 

3d.    If  P2  becomes  a  multiple  of  a  (Fig.  78),  7/,  =  0,  and  from 
(c) 

-3  =  ^  (!/22/i  -  p-)o.  +  ^{yi  +  y-dl^ 

2p 

2    "'"2^'         •  ^^ 

or,  the  subtangent  is  bisected  at  the  vertex. 


APPLICATIONS   TO   LOCI. 


189 


Also 


-3-pi  =  -fa  +  |/?-(^^^a  +  y,yS 


2p 


2p         2 


Operating  with  S  .  ttj  x 


4        4 


or,  a  perjiendicular  from  the  focus  on  the  tangent  intersects  it  on 
the  tangent  at  the  vertex. 


Fig.  TS. 


Again,  since  tt^  is  parallel  to  the  normal  at  Pi,  the  latter  maj 
be  written,  from  (<), 


whence 


or 


=  a;('-|a+|/5)  =  ^a  +  2/,/?; 


V  Vi 


a;  =  2, 


2  =  -i; 


190  QUATERNIONS. 

hence,  the  subnormal  is  constant ;  and  the  normal  is  twice  the 
pei'pendicular  on  the  tangent  from  the  focus. 
The  normal  at  Pj  may  be  written 

or 

■whence,  from  (h), 

x  =  2,     and    z'=^(!h'  +  p')  =  Tp,; 

or,  the  distance  from  (he  foot  of  the  normal  to  the  focus  equals 
the  radius  vector  to  the  point  of  contact,  or  the  distance  from  the 
2'>oint  of  contact  to  the  directrix,  or  the  distance  from  the  focus  to 
the  foot  of  the  tangent. 

The  portion  of  the  tangent  from  its  foot  to  the  point  of  con- 
tact may  be  written  za  +  pi,  in  which  z  has  just  been  found. 
Hence 

Za  +  pi=—  (?/i-  +  p")  a  +  —  (!/i-  -  p-) a  +  2/1/3, 
•2p  -Ip 

or 

the  portion  of  tlie  tangent  from  the  foot  of  the  focal  perpendicu- 
lar to  the  point  of  contact  is 

-  T3  +  pi  =  |a  -  I  /3  +  ^^(2/f  - 2r)a  +  yi/?, 
or 

-^3  +  pi  =  |La  +  |/3,  (k) 

or,  comparing  (j)  and  (k),  the  tangent  is  bisected  by  the  foccd 
perpendicular,  and  hence  the  angles  between  the  tangent  and  the 
axis  and  the  tangent  and  the  radius  vector  are  equal,  and  the 
tangent  bisects  the  angle  between  the  diameter  and  radius  vector 
to  the  point  of  contact. 


APPLICATIONS   TO   LOCI.  191 

(Tc)  is  also  the  perpendicular  from  the  focus  on  the  normal, 
and  shows  that  the  locus  of  the  foot  of  the  x)erpendicular  from  the 
focus  on  the  normcd  is  a  xKinihola,  ivhose  vertex  is  at  the  focus  of 
the  given  2^cirabola  and  ivhose  parameter  is  one-fourth  that  of  the 
given  parabola. 

88.   The  Ellipse. 

1 .  Substituting  in  the  general  equation  p  =  xa  +  yfB  the  value 
of  y  from  the  equation  of  the  ellipse  referred  to  center  and  axes 

a^y^  +  b-xr  =  a^l/, 
we  have 

p=xa  +  mi{a--xy^(3    ....     (220), 

72 

in  which  m  =  —  and  a  and  (3  are  unit  vectors  along  the  axes. 

a- 

For  unit  vectors  along  conjugate  diameters,  the  equation  of  the 

ellipse  becomes 

p  =  xa  +  m'i{a"'-x-)hft   ....     (221). 

Again,  if  4>  be  the  eccentric  angle,  the  equation  of  the  ellipse 
ma}'  be  written  in  terms  of  a  single  scalar  variable, 

p  =  cos<^  .  a  +  sin<^  . /?    ....     (222). 

2.  From  Eq.  (220)  we  have,  for  the  vector  along  the  tangent, 

1/0        o\  _i    /-J               ni         X        >3             mx  n 
a.  —  m'iicr  —  XT)    ^Xf3  =  a ———=p  =  a (d 

Vm  Va"  —  XT  y 

=  X(ya  —  mx/S)  ; 

hence,  for  the  equation  of  the  tangent  line, 

TT  =  xa  +  ?//3  +  X  (ya  -mxp)     .     .     .     ( 223)  ; 

or,  more  simply,  from  Eq.  (222),  the  vector-tangent  is 

—  sin  ^  •  a  +  cos  (fi  ,  /5, 


192  QUATERNIONS. 

aud  the  equation  of  the  tangent  is 

7r  =  cos<^  .  a  +  sin^  .  j3 -\-x(—  sin<^  .  a  +  cos<^  .  (3),  (224). 

Since  —  sin<^  .  a  +  cos<^  .  ^  is  along  tlie  tangent,  cos^  .  a  + 
sin<^  .  (i  aud  —  siu<^  .  a  +  cos<^  .  yS  are  vectors  along  conjugate 
diameters. 


89.   Examples  on  the  Ellipse. 

1.  The  area  of  the  ]}arallelogram  formed  by  tangents  draiai 
through  the  vertices  of  any  pair  of  conjugate  diameters  is  constant. 

"We  have  directl}'' 

TY[2(cos<^  .  a  +  sin^  .  (i)  2(— sin<^  .  a4-cos<^  .  /3)] 
=  4 TVa^  =  a  constant; 

namely,  the  rectangle  on  the  axes. 

2.  Tlie  sum  of  the  squares  of  conjugate  diameters  is  constant, 
and  equal  to  the  sum  of  the  squares  on  the  axes. 

For,  since  Sa^  =  0, 

(COS^  .  a  -f-  f^'incj)  .  ft)-  +{  —  s\n(f>  .  a-|-COS(^  .  /?)"  =  a-  +  /3". 

3.  The  eccentric  angles  of  the  vertices  of  conjugate  diameters 
differ  by  90? 

The  vector  tangent  at  the  extremit}'  of 

p  =  cos <f>  .  a  +  sin  ^  .  (3  (o) 

is 

—  sin  <^  .  a  +  cos  <}>  .  /3. 

This  is  also  a  vector  along  the  diameter  conjugate  to  p.  and  is 
seen  to  be  the  value  of  p  when  in  (a)  we  write  ^  +  90°  for  eft. 

4.  TJie  eccentric  angle  of  the  extremity  of  equal  conjugate  diam- 
eters is  45°  and  the  diameters  full  upon  the  diagonals  of  the 
rectangle  on  the  axes. 


APPLICATIONS   TO   LOCI.  193 

5.  Tlie  line  joining  the  X)oints  of  contact  of  tangents  is 
parallel  to  the  line  joining  the  extremities  of  parallel  diam- 
eters. 

6.  Tangents  at  right  angles  to  each  other  intersect  in  the  cir- 
cumference of  a  circle. 

7.  If  ayi  ordinate  pd  to  the  major  axis  be  produced  to  meet  the 
circumscribed  circle  in  q,  then 

QD  :  PD  :  :  a  :  6. 

8.  If  an  ordinate  pd  to  the  minor  axis  meets  the  inscribed  circle 

in  Q,  then 

QD  :  PD  :  :  &  :  a. 

9.  Any  semi-diameter  is  a  mean  proportional  between  the  dis- 
tances from  the  center  to  the  points  ichere  it  meets  the  ordinate  of 
any  point  and  the  tangent  at  that  point. 

For  the  point  p  (Fig.  82)  we  have 

p  =  cos  ^  .  a  +  sin  cfi  ,  (S. 
Also 

OT  =  a'OP  =  OQ  +  QT 

=  .t(cos^  .  a  +  sin(;6  .  /3) 

=  COS^'  .  a  +  siu</>'  .  (B  -\-t{—  siu^'  .  a  -j-  COS^'  .  (3). 

Eliminating  t,  ^ 


COS{<fi—  </)') 


But 


1 

OT  =  a;op  = ;~0P' 

cos(<^  —  </)) 


ON  =  CC'OP  =  OQ  +  QN 

z=x' (cos  (fi .  a  +  sin<^  .  /5) 

=  coscji' ,  a  +  sin</)'./3  +  t'{—  s'mcji  .a  -\-coS(f> , (3) 

Eliminating  f', 

x'  =  cos(<^  —  cj>'), 
or 

ox  =  cos(^  —  </)')op  ; 

.'.    ON  .  0T  =  OP^. 


194  QUATERNIONS. 

10.   To  Jind  the  length  of  the  curve. 

"With  the  notation  of  Ex.  G,  Art.  86,  we  obtain,  from  Eq. 
(222), 

p'=  —  sill  <ji  ,  a-\-  cos  ^  .  /?, 

Tp'=  V(a-  — 6-)sin^<^  +  62, 
s  —  So  =  I  V(a^  —  6'-')sin''^^  + 1/, 

which  involves  elliptic  functions.     If  «  =  6,  we  have,  for  the 
circle,  s  —  s^  =  |  ?•  =  r{<f>  —  <^o)  • 
From  Eq.  (220) ,  we  obtain 

p'=  a  —  m'i  (a-  —  x^)  -2.r/3, 
a 


^        a-  —  .X"         v«"  —  •1'    ^         « 

J'»-i;       a  L      e'-x- 

■         -yi — -, 
,,   sjcv^  —  x^  ^         a- 

■'o 

which  may  be  expanded  and  integrated ;  giving  for  the  enthe 
curve 

27ra   1 etc.  1, 

V        2.2      2.2.4.4      2.2.4.4.6.6  / 

a  converging  series.     If  e  =  0,  we  have,  for  the  cu'cle,  27rr. 

11.   To  find  the  area  of  the  ellipse. 
With  the  notation  of  Ex.  7,  Art.  86, 

TVpp'=  TY(cos </>  .  a  +  sin  ^  .  ft){—  sin <^  .  a  +  cos^  .  /3) 
=  TT(cos-<^  .  a/3  -  sin-</)  .  ^a)  =  TVayS  ; 

A 

or,  since  a/3  =  90°  ^_ 

if  ^TVpp'=i7ra&. 
^% 

The  whole  area  is  therefore  Trah. 


APPLICATIONS   TO  LOCI.  195 

90.   The  Hyperbola. 

1.  Let  a  and  /3  be  unit  vectors  parallel  to  the  asymptotes. 
Then,  from  the  equation, 

a2  +  62 
4 

we  have,  for  the  equation  of  the  hyperbola, 

p  =  xa  +  -/3 (225); 

or,  if  a  and  y8  are  given  vectors  parallel  to  the  asj'mptotes, 

P  =  «a+| (226); 

or,  again,  in  terms  of  the  eccentric  angle, 

p  =  SGCcji  .  a-j-tancji  ,  ^ .     .     .     .      (227). 

2.  The  equation  of  the  tangent,  obtained  as  usual,  is  from 
Eq.  (22G), 

p  =  ta-\-^  +  x(ta-(^\     .     .     .      (228), 
where  ia  —  —  is  a  vector  along  the  tangent. 


91.   Examples  on  the  Hyperbola. 

1.  7/",  ivlien  the  hyperbola  is  referred  to  its  asymptotes^  one 
diagonal  of  a  parallelogram  tchose  sides  are  the  coordinates  is 
the  radius  vector,  the  other  diagonal  is  parallel  to  the  tangent. 

If  (Fig.  79)  ex  =  ta,  xp  =^,  then 

B  R 

CP  z=ta-{-!-,      QX  =ta  —-  ; 
z  t 


196 


QUATERNIONS. 


but  ta—B  is  parallel  to  the  tangent  at  p  (Art.  90).     ta-\-^  and 
ta  —  L  are  evidcntl}'  conjugate  semi-diaractors. 

Fig.  79. 


2.  A  diameter  bisects  all  chords  parallel  to  the  tangent  at  its 
vertex. 

Let  (Fig.  79)  cp  be  the  diameter,  t  corresponding  to  the 
point  p.  The  tangent  at  p  is  parallel  to  ta—!-  and  cp  =  ta  -\-—. 
p'p"  being  the  parallel  chord, 

CP'=  CO  +  OP'=  x(ta  ^B\^yfta—B 

Also,  if  t'  correspond  to  v', 

Cv'=t'a+^', 


.-.   {x  +  y)t  =  t', 

x^-f  =  l. 


T~-F 


APPLICATIONS   TO   LOCI.  197 

Hence,  for  ever}'  point,  as  o,  determined  b}'  x^  there  are  two 
points  p'  and  p",  determined  by  the  two  corresponding  values 
of  ?/,  wliich  are  equal  with  opposite  signs. 

3.  The  tangent  at  Pj  to  the  conjugate  hyperbola  is  parallel  to 
CP  (Fig.  79). 

4.  The  p^ortion  of  the  tangent  limited  by  the  asymptotes  is 
bisected  at  the  point  of  contact. 

5.  If  from  the  point  d  (Fig.  79),  where  the  tangent  at  p  meets 
the  asymp)tote,  dn  be  drawn  parcdlel  to  the  other  asymptote^  then 
the  portion  of  pn  produced^  ivhich  is  limited  by  the  asymptotes,  is 
trisected  at  p  and  n. 

We  have  ^  ^ 

CN  =  2ta  +  xB  =  t'a+f^,  =  2ta-{--!^, 
'      '^  ^  t'  2t 

CP  =  ta+-; 
t 

,-.    PN  =  CN  —  CP  =  ia  — -^, 

2t 

and  the  equation  of  ss'  is 

''  =  '°  +  7  +  K'""2^< 

whence,  for  the  points  s,  sj 

.T  =  -l,       .T=2. 

6.  TJie  intercepts  of  the  secant  betiveen  the  hyperbola  and  its 
asymj^totes  are  equal. 

The  vector  along  the  tangent  parallel  to  the  secant  is  ta  —  -. 
Hence  (Fig.  79) 

Cr'  =Za=x(ta  +  -]  +  y(ta  —  - 
Cr"  =  z'(3=  X  (ta  +  ^\+  y'fta  -  ^\ 

•••  y  =  -y'', 

but  Op"  =  op'  (Ex.  2),  and  therefore  p"r"  =  p'rI 


198  QUATERNIONS. 

7.    If  through  any  j^oint  p"  (Fig.  79)  a  line  r"p'r'  be  drawn 
n  any  direction,  meeting  the  asymptotes  in  r"  and  b.\  then 


r'  r"  .  p'r'  =  PD 


■.'  T^"     X.'.,'       ^.^': 


8.  If  through  pj  p"  (Fig.  79)  lines  be  draicn  parallel  to  the 
asymptotes,  forming  a  parallelogram  of  which  p'p"  is  one  diagonal, 
the  other  diagonal  will  pass  through  the  center. 

The  vector  from  c  to  the  farther  extremit}'  of  the  requu-ed 
diagonal  is 

But  t"a  +  ^  is  the  ^-ector  from  c  to  the  other  extremity  of  the 
required  diagonal. 

9.  If  the  tangent  at  any  point  p  meet  the  transverse  axis  in  t, 
and  PN  he  the  ordinate  of  the  point  p  ;  then 

CT  ,  ex  =  o^, 

c  being  the  center  and  a  the  semi-transverse  axis. 
From  Eq.  (:^-7),  substituting  in  ct  =  cp  +  pt, 

X  sec  <f> .  a  =  sec  </> .  a  -f  tan  <f> .  fB  -j-  ?/  (tan  </>  sec  ^  .  a  +  sec-  <}> ,  (3)', 

1 

•'•  •'^  =  — :r-> 
sec-  <f) 

and 

CT  .  CN  =  {x  sec  ^  .  a)  (sec  </>  .  a)  =  a^, 

or 

CT  .  CN  =  a-. 

10.  If  the  tangent  at  any  point  p  meet  the  conjugate  axis  at  t', 
and  pn'  be  the  ordinate  to  the  conjugate  axis,  then 

ct'  .  cn'  =  b-, 
c  being  the  center  and  6  the  semi-conjugate  axis. 


APPLICATIONS    TO    LOCI,  199 

92.  The  preceding  examples  on  tlie  conic  sections  involve 
directly  the  Cartesian  forms.  A  method  will  now  be  briefl}' 
indicated  peculiar  to  Quaternion  analysis  and  independent  of 
these  forms. 

1.  The  general  form  of  an  equation  of  the  first  degree,  or  as 
it  ma}^  be  called  from  analogy,  a  linear  equation  in  quaternions,  is 

aqh  +  a'qb'+  a"qb"+ =  c, 

or 

^aqb  =  c,  (a) 

in  which  q  is  an  unknown  quaternion,  entering  once,  as  a  factor 

onlj',  in  each  term,  and  a,  h,  a',  h', ,  c  are  given  quaternions. 

It  may  evidently  be  written 

2Sag&  +  Vfaqb  =  Sc  +  Vc, 
whence 

2Sag&  =  Sc,  (5) 

SVagft  =  Vc.  (c) 

But 

^aqh  =  S(/6a  =  S(/S6a  +  S  .  XqYba, 
and 

Taqb  =  V(Sa  +  Va)  {Sq  +  Vg)  (Sb  +  V6) 

=  V  .  Sg(Sa  +  Va)  (S6  +  Yb) 

+  V(SaVgS&  +  SaVgV6  +  V«VgS&  +  YaYqJb) 
=  SqYab  +  V(SaS6  -  SaV&  +  S&Va)  Yq 

+  V  .  VaVgV&  +  V  .  YaYbYq  -  V  .  YaYbYq 
[Eq.  (116)]  =  SqYab  +  V(SaS&  -  SaV&  +  SbYa  -  YaYb)Yq 

+  2  VaS  .  YqYb 
=  SgVa&  +  V  .  a{Kb)Yq  +  2 VoS  .  YqYb. 

We  have  therefore,  from  (b)  and  (c), 

Sc  =  Sg2S6a  +  S  .  Vg2V6a, 

Vc  =  Sg2Va6  +  2V  .  a  (K6)  Vg  +  2  SVoS  .  VgV6, 

or,  writing 

2Sa&  =  d,     2Va&  =  S,     2V&a  =  8,'     Sg  =  i«,     Vg  =  p, 


200  QUATERNIONS. 

we  obtain 

Sc  =  tod  +  SpS; 

Yc  =  icB  +  2V  .  a{Kb)p  +  2 2VoS  .  pYb. 

"We  ma}'  now  eliminate  lo  between  these  equations,  obtaining 

Yc  .d-Sc  .8  =  d2Ya(K&)p  -  88/38'+  rf22YaS  .  pY6 

which  involves  onlv  the  vector  of  the  unknown  (luaternion  7,  and 

which,  since  Y  and  2  are  commutative,  ma^-  be  written  under 

the  general  form 

y  =  \i'p  +  2,5Sap, 

in  which  y,  a,  a,  ,  /8.  /3J are  known  vectors,  r  a  known 

quaternion,  but  p  an  unknown  vector.  This  equation  is  the 
general  form  of  a  linear  vector  equation.  The  second  member, 
being  a  linear  function  of  p,  may  be  written 

Yrp  +  2/3Sap  =  <^p  =  y     ....     (229), 

where  (ftp  designates  an}'  linear  function  of  p.  If  we  define  the 
inverse  function  (f>~^  b}'  the  equation 

<l>~\4'p)  =  P-^ 

the  determination  of  p  is  made  to  depend  upon  that  of  <^~^ 

2.  Without  entering  upon  the  solution  of  linear  equations,  it 
is  evident  on  inspection  that  the  function  ^  is  distributive  as 
regards  addition,  so  that 

<f>ip  +  p'+ )  =  </>p  +  V+     .     .     .     (230). 

Also  that,  a  being  any  scalar, 

cf>ap  =  acf>p (231), 

and 

dc}>p  =  cf,dp (232). 

3.  Furthermore,  if  we  operate  upon  the  form 

</)p  =  l/3Sap  +  \rp 


APPLICATIONS   TO   LOCI. 


201 


with  S  .  o-  X  ,  o"  being  any  vector  whatever, 

So-<^p  =  2S(o-^Sap)  +  So-(Vrp) . 
S(o-j8Sap)  =  So-|8Sap  =  SpaS^cr  =  S(paS^o-)  , 


But 
and 


S(o-Vrp)  =  S[o-V(Sr  +  Vr)p]  =  SrSo-p  +  So-(Vr)p 
=  SrSpo-  -  Sp(Vr)o-  =  S[pV(Kr)cr] . 


Hence,  if  we  designate  by  ^V, 

<^'o-  =  :SaS;8o-  +  V(Kr)o-, 

a  new  Hnear  function  differing  from  <^  by  the  interchange  of  the 
letters  a  and  f3,  and  Kr  for  ?•,  we  shall  have,  whatever  the  vectors 
p  and  (7, 

S(crc/.p)  =  S(p<^V). 

Functions,  which,  like  <^  and  <^^  enjoy  this  property',  are  called 
conjugate  functions.  The  function  ^  is  said  to  be  self-conjugate, 
that  is,  equal  to  its  conjugate  <^,  when  for  any  vectors  p,  o-, 

Scre^p  =  Spcjxr. 

93.  In  accordance  with  Boscovich's  definition,  a  conic  sec- 
tion is  the  locus  of  a  point  so  moving  that  the  ratio  of  its  dis- 
tances from  a  fixed  point  and  a  fixed  right  line  is  constant. 


1.  Let  F  (Fig.  80)  be  the  fixed  point  or 
focus,  DO  the  fixed  line  or  directrix,  and  p 

any  point  such  that  —  =  e,  the  constant  ratio 

DP 

or  eccentricity.  Draw  fo  perpendicular  to 
the  directrix,  and  let  FO  =  a,  OD=?/y,  PD  =  x'a 
and  FP  =  p.     By  definition. 


Fis-  SO. 


or 


Also 


T(pd) 

p-  =  ewa^. 

p-\-Xa  =  a-\-  yy. 


(a) 


202  QUATERNIONS. 

Operating  with  S  .  a  x  ,  we  have,  since  Say  =  0, 

Sap  +  Xa'  =  a", 
a;-a*  =  (tt2-Sa/5)2. 


or 


Substituting  in  (a) 


a'p'=e-(a'-Sapy       ....      (233), 


in  which  e  ma}*  be  less,  greater  than,  or  equal  to  unit}',  corre- 
sponding to  the  ellipse,  h3'perbola  and  parabola. 


Fig.  81. 


2.    For  the  ellipse,  Fig.  81,  putting  p  =  xa  for  the  points 
A  and  A 5  we  have 


1+e 
or,  since  p  =  xa  =  xfo, 


6  € 

X  = and     x  = ■, 


1-e 


FA  = FO, 

1+e 


FA'=  FO, 


whence 


and  therefore 


1-e 


aa'  =  2  a  =  —^ — -  FO, 
1  -  e- 


1  -e- 

FO  = a, 

e 


APPLICATIONS   TO   LOCI.  203 

which  furnish  the  well-known  properties  of  the  ellipse, 

FA  =  o(l  —  e), 
fa'=  a(l  +e), 

CF  =  ae, 

1-e 

AG  = a, 

e 

a 
CO  =-. 
e 

3.  Changing  the  origin  to  the  center  of  the  curve,  let  cf  =  a' ; 

then   cp  =  p'   and    p  =  p'—a\    a'  =  ( )  a  ;    whence 

a  =  — - —  a.     Substituting  these  values  of  p  and  a  in 

aV'  =  e'(a^-Sap)2, 

remembering  that  a'^  =  —  a^e-,  we  obtain 

aV"  +  (Sa'p')' =  -  a'(l  -  e') , 

or,  dropping  the  accents,  c  being  the  initial  point, 

a^p'  +  {Sapy  =  -a\l-e')      .     .     .    (234), 

the  equation  of  the  ellipse  in  terms  of  the  major  axis  with  the 
origin  at  the  center.  If  p  coincides  with  the  axes,  Tp  =  a  or  6, 
as  it  should. 

4.  Equation  (234)  may  be  deduced  directl}'  from  Newton's 
definition,  thus:  let  cf  =  a  (Fig.  81)  as  before,  f  and  f'  being 
the  foci,  and  cp  =  p.     Then 

FP  =  p  —  a,     f'p  :=  p  -{-  a, 
and,  by  definition, 

fp  +  f'p  =  2  a 
as  lines ;  or 

T(p-a)-|-T(p  +  a)=2a, 


204  QUATERNIONS. 

a  being  the  semi-major  axis.     AVhcnce 

V-(p  — a)-  =  2a  —  V-(p  +  a)-. 

Squaring 


-  /)-  +  2  Spa  —  a-  =  4  a-  -  4  a  V— (p+a)-  -  p2  _  2  Spa  -  a^ 

Spa  —  cr  =  —  a  V—  (p  -+-  a)-'. 
Squaring  again 

(Spa) 2  -  2  a^ Spa  +  a*  =  -  (rip"  +  2  Spa  -f-  a-) , 
a^p-  +  (Spa)^  =  —  a''  —  era-, 
or,  as  before, 

aV'  +  (Spa)-  =  -a''(l-e-"). 

94.    1 .  The  equation  of  the  ellipse 

oV  +  (Sap)-=-a''(l-e=^) 
may  be  put  under  the  form 


S. 


Cl^  p  +  aSap 


a^(l-e-)_ 
or,  in  the  notation  of  Art.  92,  writing 

cr  p  +  aSap        , 

the  equation  of  the  ellipse  becomes 

^p4>p  =  1 


(235). 


2.    B}-  inspection  of  the  value  of  <f>p  it  is  seen  that,  when  p 
coincides  with  either  axis,  p  and  (ftp  coincide. 
Operating  on  (j>p  with  S  .  o-  x  ,-we  have 

g     ,  _o^S|rp  +  SaaSap 

^'^  a\l-e-)       ' 


ArPLiCATiOiSrs  to  loci.  205 

operating  on  <^o-  = — ^ —  with  S  .  p  x  ,  we  have 

a\l— e-) 

^    ,     ^  _  ft-Spcr  +  SpaSacr  . 
^^  a\l-e')       ' 

hence 

Sp(/)o- =  So-<^p (236), 

and  ^  is  self-conjugate. 

3.  Diflferentiating  Equation  (235),  we  have 

Hclpcjip  -j-  SpcZ^p  =  0, 

Sdp<^p  +  Sp<^dp  =  0,  [Eq.  (232)] 

Sp<ji.dp  +  Sp</)cZp  =  2  Sp</)dp  =  0.  [Eq.  (236) ] 

If  TT  be  a  vector  to  any  point  of  the  tangent  hne, 

TT  =  p  +  xdp, 
whence 

Sp^(7r  — p)  =  S(7r  — p)<^p  =  0,  (a) 

or 

S7r<^p  =  Sp^p  =  Sp<^7r  =  1    .      .      .      .      (237) 

is  the  equation  of  the  tangent  line. 

From  (a)  we  see  that  c^p  is  a  vector  parallel  to  the  normal  at 
the  point  of  contact,  being  parallel  to  p  only  when  p  coincides 
with  the  axes,  as  already  remai'ked. 

4.  To  transform  the  preceding  equations  into  the  usual  Car- 
tesian forms,  let  i  be  a  unit  vector  along  ca  (Fig.  81),  and  j  a 
unit  vector  perpendicular  to  it.  If  the  coordinates  of  p  are  x 
and  ?/,  then,  since  a  =  cf, 

p  =  xi  +  yj, 


and 


_      a^p  -\-  aSap  _  _  a^(xi  +  yj)  -f  aeiS  .  aei{xi  -f-  yj) 
'f'P-~  a*(l-e-)  ~  ■  a\l-e')  T 

_      a^xi(l  —  e^)  +  a^yj 


206  QUATERNIONS. 


or,  since  1  —  e-  =  — 


=  -fM 


Sp<^p  =  1  =  -  S  .  {xi  +  yj)(^^  +  Ij, 


and 

Again,  if  x'  and  y'  are  the  coordinates  of  a  point  in  the  tangent, 

TT  =  x'i  +  y'j  ; 

and 

o?yy'+  b^xx'  =  crh^. 

The  above  applies  to  the  hyperbola  when  p  >  1,  that  is,  when 
1  —e-  = ;,  giving  the  corresponding  equations 

a^y-  —  Irx-   =  —  a-  b^, 
(i-^yy'—  b-xx'  =  —  a^b-. 

95.   Examples. 

1.    To  find  the  locus  of  the  middle  points  of  parallel  chords. 

Let  ^  be  a  vector  along  one  of  the  chords,  as  rq  (Fig.  82) , 
the  lepgth  of  the  chord  being  2?/,  and  let  y  be  the  vector  to  its 
middle  point ;  then 

p  =  y  +  ?//3    -'^iifi    p  =  y  —  y(^ 

are  vectors  to  points  of  the  ellipse,  and 

S(y+2/^)<^(y+?/^)  =  l, 
S(y-2/^)<^(y-?/^)  =  l; 

whence,  expanding,  subtracting,  and  appl3'ing  Equation  (236), 

Sy</>/3=0, 


APPLICATIONS   TO   LOCI. 


207 


the  equation  of  a  straight  line  through  the  origin.  Since  <fi/3 
is  parallel  to  the  normal  at  the  extremitj'  of  a  diameter  parallel 
to  f3,  the  locus  is  the  diameter  parallel  to  the  tangent  at  that 
point. 


Fiff.  82. 


2.    Equation  of  condition  for  conjugate  diameters. 

Denote  the  diameter  op  (Fig.  82)  of  the  preceding  problem, 
bisecting  all  chords  parallel  to  /3,  by  a.     Then 


or 


Sa<f>fi  =  0, 


In  the  latter,  (3  is  perpendicular  to  the  nomtial  ^a  at  the  ex- 
tremity of  a,  and  is  therefore  parallel  to  the  tangent  at  that 
point;  hence  this  is  the  equation  of  the  diameter  bisecting  all 
chords  parallel  to  a.  Therefore,  diameters  which  satisfy  the 
equation  Sac^/?  =  0  are  conjugate  diameters. 

3.    Supplementary  chords. 

Let  pp'  (Fig.  82)  and  dd'  be  conjugate  diameters,  and  the 
chords  PD,  pd'  be  drawn.     Then,  with  the  above  notation, 


and 


DP  =  a  —  ^, 
D'p=a-|-/S, 

S(a-f;8)0(a-/3)  =  S(a+/S)(c^a-<^^) 

=  S(ac/>a  -  acf>l3  +  ftcj>a  -  y8<^y8). 


208 
But 


QUATERNIONS. 

Sa<^a  =  1 ,      H/3<t>/3  =  1 ,      Sa<^/3  =  S/3(^a  ; 


Hence,  if  dp  is  parallel  to  a  diameter,  pd'  is  parallel  to  its 
conjugate. 

4.  Iftico  tangents  be  drawn  to  the  ellipse,  the  diameter  2'>cirallel 
to  the  chord  of  contact  and  the  diameter  through  the  intersection 
of  the  tangents  are  conjugate. 

Fig.  82. 


Let  TQ  (Fig.  82)  and  tr  he  the  tangents  at  the  extremities  of 
the  chord  parallel  to  /3,  and  or  =  tt.     Then 

OQ  =  .Ta  +  ?//?,       OR  =  Xa  ■+■  7/'/?. 

From  the  equation  of  the  tangent  Sttc^/j  =  1 ,  we  have 

S7r<^(.Ta  +  .V/S)=l, 

ii7r<l>{xa-\-y'l3)=l. 

Expanding  and  subtracting 

Sttc^/S  =  0. 

Hence,  Ex.  2,  tt  and  /3,  or  op  and  on,  are  conjugate.  The 
locus  of  T  for  parallel  chords  is  the  diameter  conjugate  to  the 
chord  through  the  center. 


APPLICATIONS    TO   LOCI.  209 

5.  If  qoq'  (Fig.  82)  he  a  diameter  and  qr  a  chord  of  contact^ 
then  is  q'r  parallel  to  ot. 

RQ  being  parallel  to  jB,  and  oq'  =  —  oq,  we  have 
RQ  =  2 yfS,     rq'  =  y(3  —  xa  —  xa  —  ?//3  ; 

whence,  directl}'  rq'  =  —  2.Ta  ;  as  also  Srq^rq'  =  0,  rq  and  rq' 
being  supplementaiy  chords. 

6.  The  points  in  ivhich  any  two  parallel  tangents  as  q'tJ  qt 
(Fig.  82)  are  intersected  by  a  third  tangent,  as  ttJ  lie  on  conju- 
gate diameters. 

The  equation  of  rt'  is  Sttc^p  =  1,  and  that  of  q't'  is  S7r'</)p'=l. 
For  the  point  x^  tt  =  tt'  ;  whence,  by  subtraction, 

S7r</)(p  —  p  )  =  0. 

7.  Chord  of  contact. 

The  equation  of  the  tangent, 

Sp<^7r  =:  1  , 

is  linear,  and  satisfied  for  both  q  and  r.     Hence,  writing  a  for  p 
as  the  variable  vector,  tt  being  constant, 

Scr^TT  =  1 

is  the  equation  of  the  chord  of  contact. 

8.  To  find  the  locus  of  x  for  all  chords  through  a  fixed  point 
(Fig.  82). 

Let  s  be  a  fixed  point  of  the  chord  rq,  so  that  os  =  o-  =  a 
constant.     Then 

Scrc^TT  :=  Stti^ct  =  1  , 

a  right  line  perpendicular  to  c^o-,  or  parallel  to  the  tangent  at  the 
extremity  of  os,  and  the  locus  of  x  for  all  chords  through  s. 


210 


QUATERXIONS. 


9.  Any  semi-diameter  is  a  mean  proportional  between  the  dis- 
tances frovi  the  center  to  the  j)oints  where  it  meets  the  ordinate  of 
any  point  and  the  tangent  at  that  point. 


OD  (Fig.  82)  and  op  being  still  represented  by  P  and  a,  let 
OT  =  x'a  and  OQ  =  p  =  a.*a  +  ?//?.  Then  from  the  equation  of  the 
tangent,  Ss-c^p  =  1 ,  we  obtain 

Sx'a<^(.Ta  +  ?/y3)=l; 

whence,  since  Sa<^/?  =  0, 

a;x''Sa^a=  1, 
or 

xx'  —  1  ; 

.*.    Xa  ,  x'a  =  a^, 
OX  .  OT  =  OP-. 


or 


10.    I/j)d'  (Fig.  82)  and  pp'  are  conjugate  diameters,  then  are 
PD  and  pd'  proportional  to  the  diameters  parallel  to  them. 

"With  the  same  notation 

DP  =  a  — ^,  D'p  =  a  +  ;3, 

vrhence 

OE  =  m  (a  —  /S)  ,       OF  =  71  (a  +  yS)  . 


and 


APPLICATIONS   TO   LOCI.  211 

From  the  equation  of  the  ellipse 

Sm  (a  -  /S)  cf>m  (a  -  /?)  =  1 ,  (a) 

S?i(a  +  ^)<^n(a+/S)=l.  (b) 

Now,  from  (a) ,  since  SyS^/?  =  SacjSa  =  1  and  S/3^a  =  Sa<^/3  =  0, 
2m-=l. 
29t-=l; 


Similarly,  from  (b), 


Also 

DP  :  d'p  : :  T(a  -  ^)  :  T(a  +  ^) 

::T??i(a  — /3)  :  Tii(a  +  /3) 
:  :  OE  :  OF. 

1 1 .  TJie  diameters  along  the  diagonals  of  the  parallelogram  on 
the  axes  are  conjugate;  and  the  same  is  true  of  diameters  along 
the  diagonals  of  any  jKirallelogram  lohose  sides  are  the  tangents  at 
the  extremities  of  conjugate  diameters. 

12.  Diameters  parallel  to  the  sides  of  an  inscribed  parallelo- 
gram are  conjugate. 

Fiff.  S3. 

K 


Let  the  sides  of  the  parallelogram  (Fig.  83)  be 
pp'  =  a,     PQ  =  /?, 


and  let 
Then 


OP 


OP  =  p,      OQ  =  p. 
'-p+a,      OQ'  =  p'  +  a,      p'-p  =  ff. 


212  QUATERNIONS. 

From  the  equation  of  the  ellipse,  S/3<^p=  1 ,  we  have  for  q'  and  p' 

S(p'+a)<^(p'+a)=l, 
S(p+a)<^(p4-a)=l; 

whence,  since  Sp4>p  =  Sp'  4>p  =  1 , 

2Sa<^p'+Sa<;f)a  =  0, 
2  Sa<i>p  +  Sa^a  =  0. 

S/3  <^a  —  S/j^a  =  0, 


Subtracting 


or 


13.  The  rectangle  of  the  perpendiculars  from  the  foci  on  the 
tangent  is  constant,  and  equal  to  the  square  of  the  semi-conjugate 
axis. 

Fi-.  S3. 


Let  the  tangent  be  drawn  at  r  (Fig.  83)  and  or  =  p.  Tlien 
<}>p  is  parallel  to  the  normal  at  r,  that  is,  to  the  perpendiculars 
FD,  f'd!     Hence,  of  being  a, 

OD  ^  X  (j>p  —  a, 
OD  =  a  +  Xc^tp, 

which,  since  d  and  d'  are  on  the  tangent,  in  Sttc^/d  =  1  give 
S(.7;  <^p  —  a)  <^p  =  1 , 

S(a  +  Xcf)p)(l>p=  1, 

or 

X'(cf>py=\  -hSacf^p, 
X  (</)p)"  =1  —  Sa^p  ; 


APPLICATIONS   TO  LOCI. 


213 


whence 


and 


Hx^p   =FD    =T 


1  +  Sa<^p 

(jip 
1  —  Ha(j>p 

4>p 


FD  X  f'd'  =  T 


f,,f_rrl  -(Sac^p)^ 


But 


iM' 


f      0?p  +  aSapY_  a^cv^p-)  +  2  ff^(Sap)^  +  a^Sap)^ 


or,  substituting  a'p'  from  Equation  (23-4)  and  a-  =  — a^e^, 

_  (Sap)2-fl* 

Also 

l-(Sa<^p)2=l 


ft"  Sap  +  a' Sap 


a*-(Sap)^ 


•  •.  FD  X  f'd'  =  T 


ft^l  -e-) 

ft^-(Sap)-  ft'^(l-e^) 

a''  (Sap)-  — a 


-^=a-(l~e2)  =  52 


14.    The  foot  of  the  perpendicular  from  the  focus  on  the  tangent 
is  in  the  circumference  of  the  circle  described  on  the  major  axis. 

To  prove  this  we  have  to  show  that  the  line  od  (Fig.  83)  is 
equal  to  a.     Now 

CD  =  a  +  X<jip 

4>p{\—^a4p) 

="+    (#)= 

from  the  preceding  example.     Hence 

2Sa<^p(l  -Sa<^p)    ,    (1-Sa(^p)2 


(oD)-  =  a2  4-: 


{^pY 


=  —  a^e^  —  a^(l  —  e-)  =  — a^ ; 


{i>py 

ft^-(Sa.p)^ft'^(l-e') 

a*         (Sap)^— a* 


214  QUATERNIONS. 

The  Parabola. 

96.   1.    Resuming   Equation    (233)    and   making   e  =  l,   the 
equation  of  tlie  parabola  is 

a- p- =  {a- -  SapY (238), 

which  ma}'  be  written 

p2  +  2Sap-a-nSap)^_, 

'>  —  '■J 

a" 

or 

„      p  +  2  a  —  a~^aSap~l 

Sp    ^ , L    =1; 

_  a 

in  which,  if  we  put 

p  —  a~'  Sap 

9P  =  2 ' 

a 

we  have  for  the  equation  of  the  paral)ola 

Sp(</>p  + 2  «-')=! (239), 

and,  as  in  the  case  of  the  elHpse, 

S(T(f>p  =  Sp(t>ar (240) . 

Operating  on  <^p  b}^  S  .  a  x  ,  we  obtain 

Sa<^p  =  0 (241); 

hence,  (ftp  is  a  perpendicular  to  the  axis. 
Operating  on  ^p  b}'  S  .  p  X 

Sp<^p  =  e^I1^4^^'=a^(<^p)^.     .     .     (242). 

a 

2.    Ditforentiatiug  Equation  (239),  we  have 

2Sp<^dp  +  2Sf?pa-i  =  0. 
For  an}'  point  of  the  tangent  Hue  to  which  the  vector  is  tt, 
~  —  p  +  xdp, 


APPLICATIONS   TO   LOCI.  215 

from  which,  substituting  dp  in  the  above, 

Sp</>(7r-p)      +S(7r-p)a-i  =  0, 

S(p</)7r  —  p</)p  +  7ra~'— /3tt"^)  =  0  ;  (a) 

or,  since  Sp^p  =  1  —  2Spa~^  [Eq.  (239)], 

Sttc^P  —  1+2  Spa~^  +  S7ra~^  —  Spa~^  =  0, 

whence 

S7r(</.p  +  a-i)  +  Spa-'=  1  .       .       .      .      (243), 

the  equation  of  tlie  tangent  line. 

3.  From  (a)  we  obtain 

S(,r-p)(c/.p  +  a-i)  =  0;         • 

or,  since  tt  —  p  is  a  vector  along  the  tangent, 

4>p  +  a~^ 
is  in  the  direction  of  the  normcd. 

4.  If  o-  be  a  vector  to  an}'  point  of  tlie  normal,  the  equation 
of  the  normal  will  be 

a  =  p  +  X(cf,p  +  a-') (244). 

5.  The  Cartesian   form   of  Equation  (239)   is  obtained   b}' 
making 

p  =  xi  +  yj,     a  =  FO  (Fig.  80)  =  -jn  ; 

xpi 

^i  +  yJ 


whence,  Equation  (239)  becomes 

yi-zl=i- 

F       P 

•••  y-  =  '2px-\-i?, 
the  equation  of  the  parabola  referred  to  the  focus. 


216  QUATERNIONS. 

97.   Examples. 

1.    Tlte  tiublaityent  is  bisected  at  (he  vertex. 

rig.  SI. 


\ 

\ 

^ 

^\ 

\ 

\ 

^^ 

-^x 

/       '■        \ 

^ 

y 

/ 

/  1  \ 

j 

j 

! 

i 

\ 

\ 

T 

() 

A 

V 

1 

\ 

.M 

N 

gives 


(«) 


We  have  (Fig.  84)  ft  =  .ra,  which  in  the  equation  of  the  tangent 

S7r(<^p  +  a-^)  +  Spa-i=l 
Sxa(<f>p  +  a~^)  +  Sa~*/3  =  1. 

But  Sa<^/3  =  0  ;  hence 

rc  +  Sa-V=  1  ; 
multipl3'ing  by  a 

SCa  +  aStt"  p  =:  a, 
(.r  —  i)a  =  a  —  ^-a  —  aSa~'p 

=  ^a  — aSa"Vi 
AT  =  —  AF  —  aSa"'p. 

But  the  value  of  <^/j  gives 

a*  ^/j  =  p  —  a"  Sap  ; 


APPLICATIONS   TO   LOCI.  217 

and,  since  ^p  is  a  vector  along  bip  and  a~^^ap  a  vector  along  fm, 
from  p  =  FM  +  MP  we  have 

FM  =  a~'  Sap  =  aSa~^p,  (6) 

-  MP  =  a^(jip;  (c) 

.'.    AT  =  —  AF  —  FJI  =  —  AM, 

or,  as  lines, 

AT  =  AM. 

2.    The  distances  from  the  focus  to  the  point  of  contact  and  the 
intersection  of  the  tangent  ivith  the  axis  are  equal. 


or  (Fig.  84), 


a;a  =  a  —  aSa  ^p, 

(FT)2  =  (a-aSa-V)' 
=  (a-a-^Sap)2 

(a^  —  Sap)^ 


[Eq.(238)]  =p^ 

.'.    FP  =  FT. 

3.  The  siiJmormal  is  constant  and  equcd  to  hcdf  the  parameter. 

The  A^ector-normal   being    (/)p  +  a~^    (Art.   96,   3),   we   have 
(Fig.  84) 

PN  =  2(^p+a  ^); 
but 

PN  =  PM  +  JIN 

=  -  a-<^p  +  xa  ;  [Ex.  1,  (e)  ] 

Z  =■  —  a"  =  X'a', 

or 

X'  =  —  1 ,      Xa=-  —  a  ; 

or,  the  distances  mn  and  fo  are  equal,  and  the  subnormal  =  p., 
a  constant. 

4.  The  perpendicular  from  the  focus  on  the  tangent  intersects 
it  on  the  tangent  at  the  vertex,  and  aq  =  ^mp  (Fig.  84). 


218 


QUATERNIONS. 


Since  (Ex.  2)  fp  =  ft  =  rn,  fd  is  perpciulicular  to  rr  or  par- 
allel to  px.     Otherwise  : 

NP  =  -  2;(<^a  +  a-')  =  a-(^p  +  a"')  (Ex.  3) 

=  a-(f)p  4-  a  =  MP  +  FO,  [Ex.   1,  (c)] 

.  But 

^  FI)  =  FQ  =  Tf  (i'<f>p  +  ^  a 
=  ^  d-cf>p  +  FA  ; 
.'.    ^a-</>p  =  AQ  =  ^MP. 

5.  To  find  the  locus  of  the.  intersection  of  the  perpejicUcukw 
from  the  vertex  on  the  tangent  and  the  diameter  ^yroduced 
through  the  x>oint  of  contact. 


Fig.  84. 


Let  Fs  =  C7  (Fig.  84)  be  a  vector  to  a  point  of  the  locus. 
Then 

FS  =  FA  +  AS  =  FP  -f  PS, 
<j  =  \ci.  +  Z{(f>p  -\-  a-^)  =  p  +  Xa. 


APPLICATIONS    TO   LOCI.  219 

Operating  with  x  S  .  cf)p.  tlien,  since  Sa^p  =  0  [Eq.  (241)], 

z{cf>py-  =  Sp<^p  =  a^(0p)^ ;  [Eq.  (242)] 

.•.    Z  ^  a", 

and 

(r=  ^a  +  ct"((^p  +  a~')  =  l^a  +  orcjip, 

ov 

Operating  with  x  S  .  a 

S(cr  —  |^a)a=  0, 

So■a  =  -f(Ta)^ 

or  [Eq.  (180)],  the  locus  is  a  right  line  perpendicular  to  the 
axis  and  fp  distant  from  the  focus. 

6.    To  find  the  locus  of  the  intersection  of  the  tangent  and  the 
perpendicidar  from  the  vertex. 

If  the  origin  be  taken  at  the  vertex,  then  since  ^p  +  a~'  is  a 
vector  along  the  normal,  the  equation  of  the  locus  will  be 

7r  =  .r(</)p  +  a-^).  (a) 

To  eliminate  .t,  operate  with  S  .  a  x  which  gives 

X  =  SttTT,     whence     Sa"  V  =  —  '—. 

d- 

To  eliminate  p,  the  equation  of  the  tangent,  S7r(^p  +  a~^)  + 
Spa~^  =  1 ,  for  the  new  origin  becomes 

S('^  +  ^V<^p  +  a-')-f  Spa-i=l, 

or 

2  Sttc^p  +  2  Sa- V  +  2  Sa-V  =  1  • 

Operating  on  (o)  with  x  S  .  ^p,  whence  Stt^p  =  .r(<^p)-,  the 
precedmg  equation  becomes 

2x-(<^p)^-^  +  2Sa-V=l.  (6) 


220  QUATERNIONS. 

Also  [Eq.  (212)]  Sp4)p  =  a'((f)p)-,  which,  in  the  equation  of 
the  parabola  Sp(^/3  +  2a~')  =  1,  gives 

a«(<^p)='  +  2Sa-V  =  l.  (C) 

"Whence,  from  {h)  and  (c),  by  subtraction, 

z  crx  -\-  tr 
But,  from  (a), 

.  ,    X,       TT- —  2 X^Tra'^ -j- X^ a~-       --    ,    1 

(<^p)"  = Z =  -  +  — • 

X-  XT        «- 

Equating  these  values  of  (<^p)",  and  substituting  the  value  of  .r, 

27r2Sa7r  -  f^-rr  +  (SaTr)^  =  0, 

■which  is  the  equation  of  the  locus  required.     To  transform  to 
Cartesian  coordinates,  make 

TT  =  xi  +  yj,     and     a  =  ai, 
whence 

tt  =  —  (x-  +  y-) ,     SttTT  =  —  ax,     a-  =  —  a-, 
and 

r  = ' 

(( 

X 

2 

the  equation  of  the  cissoid  to  the  circle  whose  diameter  is  the 
distance  from  the  vertex  to  the  directrix. 

7t  If  pp'  (Fig.   75)   be  a  focal  chord,  and  pa,  pa'  produced 
meet  the  directrix  in  d',  d,  then  icill  pd  and  v'd'  be  parallel  to  af. 

Ad'=  —  .TAP  =  AO  +  01),' 

or 


(a  \        a    , 

72-p)  =  7^  +  yy 


Operating  with  S  .  a  x 

X{a?  —  2  Sap)  =  a?.  (a) 


APPLICATIONS   TO   LOCI. 


221 


Now  FP  =  p  and  fp'  =  —  x'fj  are  vectors  to  points  on  the 
curve,  and  hence  satisfy  its  equa- 
tion.    Whence  [Eq.  (238)]  rig.  75  (6i.). 

a.^p~  ^  (a^  —  Sap)-, 
x'^ay  =  {a'  +  x'Sapy-; 

.-.  x'\a?  -  SapY  =  (a2+  x'SapY  ; 
or 

a;'(a-  —  Sap)  =  a"  +  x'Sap, 
.-.   a;'(a- —  2Sap)  =  a". 

Hence,  comparing  witli  (a), 

X  =  x', 

or,  the  sides  produced  of  the 
triangle  apf  are  cut  propor- 
tionately, and  therefore  d'p'  is 
parallel  to  af. 

8.  If,  icith  a  diameter  equal  to  three  times  the  focal  distance, 
a  circle  be  described  icith  its  center  at  the  vertex,  the  common 
chord  bisects  the  line  joining  the  focus  and  vertex. 

The  equation  of  the  curve  being 

a-p-  =  (a-  —  Sap)  ',  (a) 

that  of  the  circle  whose  center  is  a  (Fig.  75),  i-eferred  to  f,  is 
of  the  form  [Eq.  (210)] 

T(p-y)    =TfS, 
or,  b}'  condition, 


T 


Tfa; 


which,  in  (a),  gives 
which  is  the  proposition. 


p-  =  Sap  -h  Y%  a-. 


Sap  =  --, 
4 


222  QUATERNIONS. 


9a  The  Cycloid. 


1.  Let  a  and  (i  be  vectors  along  the  liase  and  axis  of  the 
cycloid  and  T/?  =  Ta  =  r,  the  radius  of  the  generating  circle. 
Then,  for  any  point  p  of  the  curve, 

x=rd  —  r  sin  0  =  r($  —  sin  6) , 
y  =  r   —  r  cos  ^  =>•(!— cos^), 

and  the  equation  of  the  c^x-loid  is 

p  =  {e-  sin^)a  +  (1  -  cose)(3. 

2.  The  vector  along  the  tangent  is 

(1— cose)a  +  sin6  .  /?, 
and  the  equation  of  the  tangent  is 

Tr  =  (e-  sin^)a  +  (1  -  cos^)/?  +  ^[(1-  cos^)a  +  sin^  . y?]. 

3.  The  vector  from  r  to  the  lower  extremit}-  of  the  vertical 
diameter  of  the  generating  circle  through  p  is 

PC  =  —  (\  —  cos6)f3  +  sm6  .  a, 

and,  from  the  above  expression,  for  the  vector-tangent  pt, 

S(pc  .  pt)  =  0  ; 

hence  pc  is  perpendicular  to  the  tangent,  or  the  normal  passes 
through  the  foot  of  the  vertical  diameter  of  the  generating  cir- 
cle for  the  point  to  which  the  normal  is  drawn,  and  the  tangent 
passes  through  the  other  extremity. 

4.  If,  through  p,  a  line  be  drawn  i)arallol  to  the  base, 
intersecting  the  central  generating  circle  in  q,  show  that 
PQ  =  r(7r  —  ^)  =  arcQA,  a  being  the  upper  extremitj-  of  the 
axis. 


APPLICATIONS    TO   LOCI.  223 

5.  TTith  the  notation  of  Ex.  6,  Art.  86, 

p'  =  (1  —  cos  6)a+  sin  6  .  /?, 
p'2  =  _  [  ( 1  -  COS  6) '  +  sin-  ^]  r, 

Tp'  =  r  Vl  —  2  cos  6  +  cos-  d  +  sin-  (9  =  ?-V2  — 2cos^ 

=  2rsin^e; 

/•"        "  0 

s  —  So  =  I  "2 r sin|^  =  [4 r cos|^]  _  =  8 r, 

the  length  of  the  entire  cnrve. 

6.  With  the  notation  of  Ex.  7,  Art.  86, 

TTpp'=  TV[  (^  -  sin  0) sin ^  .  a^S  +  (1  -  cos Oy-jSa'] 
=  TY[(^  sin^  -  sin-^  -(1-  cos^)-]a^ 
=  9-(^sin^  +  2cos^-2). 
A  -Ao  =  7-2  fie  sin  61  +  2  cos  ^  -  2) 

=  r^(sin^-^cos^  +  2sin^-2^) 


^(3siu^-^cos^-2^) 
the  whole  area  of  the  cun^e. 


=  3  Trt-^, 

2  7T- 


99.   Elementary  Applications  to  Mechanics. 

1.  If  6  be  the  magnitude  of  any  force  acting  in  a  known  di- 
rection, the  force,  as  having  magnitude  and  direction,  may  be 
represented  bj'  the  vector  symbol  /3,  which  is  independent  of 
the  point  of  application  of  the  force.  In  order,  completely,  to 
define  the  force  with  reference  to  an}'  origin  o,  the  A-ector  OA=a, 
to  its  point  of  application  a,  must  also  be  given.  For  concur- 
ring forces,  whose  magnitudes  are  b',  b','  ,  we  have,  for  the 

resultant,  (3  =  2y3,'  which  is  true,  whether  the  forces  are  compla- 
nar  or  not,  and  is  the  theorem  of  the  polygon  of  forces  extended. 
For  two  forces,  /S  =  /?'+  /3"  ;  whence  (S'  =  /3'-  +  /S"-  -f  2  S/S'^",  or 


224  QUATERNIONS. 

h^  =  &'2  ^_  7/-2  ^  2Z;'i"  COS 5,  which  is  the  theorem  of  the  parallelo- 
gram  of  force  a.  For  an}'  number  of  concurring  forces,  the  con- 
dition of  c'(iuilil)riuin  will  be  1(3'=  0.  For  a  particle  constrained 
to  move  on  a  phine  curve  whose  equation  is  p=z  (f){^t),  dp  being 
in  the  direction  of  tlie  tangent,  since  the  resultant  of  the  eA 
traneous  forces  must  be  normal  to  the  curve  for  equilibrium,  we 
have 

Sdp1/3'=Sdp/3  =  0.  (a) 

2.  If  OA'=a;  and  (3'  is  a  force  acting  at  \',  then  T\a'/3'=a'b'  sin^ 
is  the  numerical  value  of  the  moment  of  the  couple  f3'  at  a'  and 
— /3'  at  o.  Kepresenting,  as  usual,  the  couple  bv  its  axis,  its 
vector  s^mViol  will  be  \a'/3'.  If  — /S'  act  at  some  point  other 
than  the  origin,  as  c'  and  oc'=  y',  the  couple  will  be  denoted  by 
V(a'— y')/?.'  From  this  vector  representation  of  couples,  it  fol- 
lows that  their  comjjosition  is  a  process  of  vector  addition;  hence 
the  ref^idtant  coiqyle  is  2V(a'— y')y3J  and,  for  equilibrium, 
2V(a'— y ')/?'=  0.  If  the  couples  are  in  the  same  or  parallel 
planes,  their  axes  are  parallel  and  T2  =  2T.  Since  a!—y'  is 
independent  of  the  origin,  the  moment  of  the  couple  is  the  same 
for  all  points.  Since  V(a'—  y')/S'=  Va'/3'— Vy'/?^  the  moment  of 
a  coxiple  is  the  cdrjehraic  sum  of  the  moments  of  its  component 
forces.  If  the  forces  are  concurring,  and  a'  is  the  vector  to 
their  common  point  of  application,  2Va'/S'=  y2a'^'=  Va'2/3'  = 
Va'^,  or  the  moment  of  the  restdtant  about  any  point  is  the  sum 
of  the  moments  of  the  component  forces.  "When  the  origin  is  on 
the  resultant,  a'  coincides  with  /?'  in  direction,  and  Ta'/S  =  0  ;  or 
the  cdgehraic  sum  of  the  moments  about  any  point  of  the  resultant 
is  zero.  If  a  single  force  /3'  acts  at  a(  we  ma}-,  as  usual,  intro- 
duce two  equal  and  opposite  forces  at  the  origin,  or  at  any  other 
point  c'  and  thus  replace  /S'^.  by  /?'o  and  Ma'f^',  or  by  /J'^.  and 
V(a'—  y')(S'.  If  ^  be  a  unit  vector  along  an)'  axis  oz  through  the 
origin,  then  the  moment  of  (3'  acting  at  \',  with  reference  to  the 
axis  oz,  will  be  -  S/8'a'^,  or  -  S  .  O'^S'^:  If  /S'  and  C  are  in  the 
same  plane,  in  which  case  they  either  intersect  or  are  parallel ; 
or,  if  the  axis  passes  through  a'  there  will  be  no  moment:  in 
these  cases,  a',  /3'  and  C  are  complanar,  and  —  S/3'a'{  =  0. 


APPLICATIONS   TO   LOCI.  225 

3.  If  the  forces  are  parallel,  theli-  resultant  /?  =  S/3'=  2&'Uy8' 
=  U/326' ;  and,  therefore,  for  equilibrium,  2T/3'=  %h'=  0.  The 
moment  of  a  force  with  reference  to  any  axis  oz  through  the 
origin  being  —  S^S'a'^,  and  the  moment  of  the  resultant  being 
equal  to  the  sum  of  the  moments  of  the  components,  we  have 
Sy8aC=2Sy8'a'^,  which,  for  parallel  forces,  becomes  S(2&' .  U/?  .  a^) 
=  S(U;826'a' .  ^),  which,  being  true  for  an}-  axis,  is  satisfied 
for  2&' .  a  =  -^b'a' ; 

25'a' 


26' 


(&) 


which  is  independent  of  U;8,  and  hence  is  the  vector  to  the  cen- 
ter of  ixtrallel  forces.  When  26'=  0,  the  abote  equations  give 
/3  =  0  and  a  =  oo,  the  sj^stem  reducing  to  a  couple.  For  a  sys- 
tem of  particles  whose  weights  are  to',  ^v',' ,  we  have  the  vec- 

^    f  t 
tor  to  the  center  of  gravity  a  = —.     From  this  equation, 

2^o' 
2^o'(a  —  a')  =  0  ;  whence,  if  the  particles  are  equal,  the  smn  of 
the  vectors  from  the  center  of  gravity  to  each  particle  is  zero  ;  and, 
if  unequal,  and  the  length  of  each  vector  is  increased  propor- 
tionatety  to  the  weight  of  each  particle,  their  sum  is  zero.     For 

?o'2a' 

equal  particles,  a  = -,  or  the  center  of  gravity  of  a  system  of 

'2,10' 

equal  x)o,rticles  is  the  mean  point  (Art.  18)  of  the  polyedron  of 
tvhich  the  pxirticles  are  the  vertices.  For  a  continuous  body 
whose  weight  is  20,  volume  i',  and  density  d  at  the  extremity  of 

a,  a  = ,  in  which  2  may  be  replaced  b}'  the  integral  sign 

if  the  density'  is  a  known  function  of  the  volume.  For  a  homo- 
geneous body,  a  = ,  which  is  applicable  to  lines,  surfaces 

'^clv 

or  solids,  v  representing  a  line,  area  or  volume.  Thus,  for  a 
plane  cm've  p=  (^{f)  =  a,'  civ  =  els  =  Idp  =  Tcf)'(t)dt  and 


'-!—^ _.  (c) 

CTcl>'{t)dt 


226  QUATERNIONS. 

4.    General  conditions  of  equilihrium  of  a  solid  hodij.     Lot 

the  forces  ft',  /?'' ,  act  at  the  points  a,'  a" of  a  solid  body, 

and  oa' =  aj  OA"  =  a," Replacing  cacli  force  l)y  an  equal 

one  at  the  origin  and  a  couple,  the  given  s3-steni  will  be  equiva- 
lent to  a  sjstem  of  concurring  forces  at  the  origin  and  a  S3stem 
of  couples.     Hence,  for  equilibrium, 

2/3'=  0,  (d) 

2ya'/3'=0.  (e) 

Let   i  be   the   vector   to   any   point   x.      Then,   from    (rf), 
V  .  ^2/8'=  0,  and  therefore,  from  (e),  V  .  ^2/3'=  SVa'/S' ;  whence 

2V/5'a'-  SY/S't^  =  2y/3'(a'-  c?)  =  0.  (/) 

Converseh',  |  being  a  vector  to  any  point,  the  resultant  couple, 
for  equilibrium,  is  2y(a'-  ^)/3'=  0  ;  .-.  5ya'^'=  (»  and  1(3'=  0. 
Therefore  (/)  is  the  necessary  and  suliicient  condition  of  equi- 
librium. 

This  condition  mav  be  othem-ise  expressed  by  the  principle 

of  virtual  moments.      Let  8'  h" be  the  displacements.     Tiien 

the  virtual  moment  of  (i'  is  —  S,8'3' ;  and,  for  equilibrium, 
2S/3'S'=().  This  equation  involves  ((Z)  and  (e).  Thus,  if  the 
displacement  corresponds  to  a  simple  translation,  S'=8"=8"' 
=  etc.  =  a  constant,  and  we  maj-  write  2S^'5'  =  SS2/3'  =  0  ; 
whence,  since  8  is  real,  2/3'=  0.  Again,  if  the  displacement 
corrcsi)onds  to  a  rotation  about  an  axis  ^,  C  being  a  unit  vector 
along  the  axis, 

a'=  tHa'  =  t\»W+  na')  =  -  ^W -  ^^1 

the  last  term  being  a  vector  perpendicular  to  the  axis.  For  a 
rotation  about  this  axis  through  an  angle  0,  this  term  becomes 
—  C^  C^ia'=  —  C cos^  y^a'+  sin^  V^aJ  and  a'  becomes 

a'l  =  -  CSCa' -  ^  COS  ^  \Ca'  +  sin  6  YCal 
which,  for  an  infmiteh'  small  displacement, 


APPLICATIONS    TO   LOCI.  227 

Placing  the  scalar  factor  under  the  vector  sign  and  writing  ^ 
simply  for  Ot,,  to  denote  the  indefinitely  short  vector  along  oz, 

a'+8'=a'+\V; 

or,  8'=  YCa!     Hence  2Sy8'8'=  SS/3'VCa'=  S^2Va'/3' ;  or,  since  C  is 
not  zero,  2Ya'^'=  0. 

5.    Illustrations. 

(1)  Three  concurrent  forces,  represented  in  magnitude  and 
direction  by  the  medials  of  any  triangle,  are  in  equilibrium. 
(See  Ex.  2,  Art.  17.) 

(2)  If  three  concurring  forces  are  in  equilibrium,  they  are 
complanar.  By  condition,  jS' -\-  /3"+  (3'"— 0.  Operating  with 
S  .  /S'/5"x  ,  we  have  S/3'/S"/5"'=  0. 

(3)  In  the  preceding  case,  operating  with  V.  /S'x,  we  have 
V/3'/3"+V/?'/?"'=  0 ;  whence,  since  the  forces  are  complanar, 
TV/37i"=-  TY f3'l3',"  or  b'b"  sin(y8;  /3")  =  bV"  sm{/3',  ft'") .  A  sim- 
ilar relation  may  be  found  for  an}'  two  of  the  forces  ;  whence 

b':b":b"':  :  sin(/3;'  /5"'):  sm{f3',  /3"'):  sin  (ft',  ft"). 

(4)  If  two  forces  are  represented  in  magnitude  and  position 
b}'  two  chords  of  a  semicircle  drawn  from  a  point  on  the  circum- 
fei'ence,  the  diameter  through  the  point  represents  the  resultant. 

(5)  A  weight,  iv',  rests  on  the  arc  of  a  vertical  plane  curve, 
and  is  connected,  by  a  cord  passing  over  a  pulley,  with  another 
weight,  ivi'  Find  the  relation  between  the  weights  for  equili- 
brium. 

(a)  Let  the  curve  be  a  parabola,  and  the  pulley  at  the  focus. 
Then,  from  Eq.  (a)  of  this  article,  the  equation  of  the  curve  be- 
ing p  =  —  (y--2r)a  +  >jft,  we  have 


228  QUATERNIONS. 

ill  which  r  =  radius  vector.     Hence 

w  ?/  X  y 

p         pr  r  ' 

or,  since  r  =  rc  +  ^),  iv'=w'.'     Ilcnce,  if  the  weights  are  equal, 
equilibrium  will  exist  at  all  points  of  the  curve. 

(b)  Let  the  curve  be  a  circle  and  the  pullev  at  a  distance  m 
from  the  curve  on  the  vertical  diameter  produced.  "With  the 
origin  at  the  highest  point  of  the  circle,  p  =  xa  -f  '^'2iix  —  a:^(i. 
Hence,  r  being  the  distance  of  the  pulley  from  iv', 


<'t-'^+°)  ('"'"-' 


r'>^. 


riv' 
K  +  m' 


(c)  Let  w  be  placed  on  the  concave  arc  of  a  vertical  circle, 
and  acted  upon  b}"  a  repulsive  force  varying  inversely  as  the 
square  of  the  distance  from  the  lowest  point  of  the  circle.  To 
find  the  position  of  equilibrium.  The  origin  being  at  the  lowest 
point  of  the  circle,  and  r  the  distance  required,  let  ])  be  the 

intensity  of  the  force  at  a  unit's  distance ;  then  4  will  be  its 
intensit}'  for  any  distance  ?•,  and 

^  /     ,  n  —  x   \  fxa  +  ?//?  p  \      n 

whence  , — 

r  =  \i — 

((?)  Let  w^  rest  on  a  right  line  inclined  at  an  angle  b  to  the 
horizontal,  and  connected  with  h;"  by  a  cord  passing  over  a  pul- 
ley at  the  upper  end  of  the  line.  Find  the  relation  between  the 
weights.  With  the  origin  at  the  lower  end  of  the  line,  its  equa- 
tion is  p  =  xa.  If  (3  is  in  the  direction  of  iv',  then  Sa(i(j'/3-)-M;"a) 
=  0;     .'.   ic"=io'iiinO. 

(6)  To  find  the  center  of  gravity  of  three  equal  particles  at 
the  vertices  of  a  triangle,     a,  b,  c  being  the  vertices,  the  vector 


APPLICATIONS   TO   LOCI.  229 

from  A  to  the  center  of  gravity  of  the  weights  at  A  and  b  is 
^AB  =  AD.  The  vector  to  the  center  of  gravit}'  of  the  three 
weights  is  ■J(ab  +  ac)  =  |-ab -(-a;DC  =  ^ab +  a;(  — |^ab  +  ac)  ; 
.'.  0;=^,  and  the  required  point  is  the  center  of  gravity  of  the 
triangle. 

(7)  Find  the  center  of  gravit}'  of  the  perimeter  of  a  triangle. 

(8)  Find  the  center  of  gravit}'  of  four  equal  particles  at  the 
vertices  of  a  tetraedrou. 

(9)  Show  that  the  center  of  gi'avity  of  four  equal  particles 
at  the  angular  points  of  an}'  quadrilateral  is  at  the  middle  point 
of  the  line  joining  the  middle  points  of  a  pair  of  opposite  sides. 

(10)  The  center  of  gravity  of  the  triangle  formed  b}'  joining 
the  extremities  of  perpendiculars,  erected  outwards,  at  the  mid- 
dle points  of  au}'  triangle,  and  proportional  to  the  corresponding 
sides,  coincides  with  that  of  the  original  triangle.  Let  abc  be 
the  triangle,  bc  =  2  a,  ca  =  2/S  and  e  a  vector  perpendicular  to 
the  plane  of  the  triangle.  Then,  if  m  is  the  given  ratio,  b  the 
initial  point,  and  Ri,  r^,  Rg'the  extremities  of  the  perpendiculars 
to  BC,  CA,  AB,  respectivel}', 

BRi  =  a  +  mea,     BR2  =  2  a  +  ^  +  me/3,     BRg  =  a -\-  (3  —  me  (a  +/S)  ; 
.-.    i(BRi  +  BRo  +  BR3)  =  i(4a+2/?)  =  i[2a+2(a  +  /3)]. 

(11)  To  find  the  center  of  gravity  of  a  circular  arc.  The 
equation  of  the  circle  p  =  ?-(cos^  .  a  +  sin^  .  ^8),  gives  clp  = 
r{—sine  .  a  +  cos^  .  ^)d9; 

Ccl>{e)Tcf,'(0)cl6       Cr(cose  .  a  +  sin^  .  ft)d9 


CT<f>'(6)cl6  Ccie 


For  an  arc  of  90°    integi-ating  between  the  limits   -    and  0, 
tti  =  — (a  +  /3) ,  the  distance  from  the  center  being  —  V2  ;  which 


230  QUATERNIONS. 

may  be  obtained  directl}'  also  by  integrating  between  the  limits 
-  and  —  -.  For  a  semicircumfereucc  or  arc  of  00°  we  have,  in 
like  manner,  —  and  — . 

(12)  If  a,  y8,  y  are  the  vector  edges  of  any  tetraedron,  the 
origin  being  at  the  vertex,  then  p  —  a,  /S  —  y,  a  —  /?  ai'e  hnes  of 
the  base,  p  being  an}'  vector  to  its  plane.  Hence  this  plane  is 
represented  by  S  (p  —  a)  (/J  —  y)  (a  —  ^)  =  0  ;  .-.  Sp  {\afS  + 
Vya  +  V/iy)  —  Huf^y  =  U.  If  8  be  the  vector  perpendicular  on 
the  base, 

8  =  x(Yal3  +  Vya  +  Y^y)  =  ..  ,,  J"^^^—^^ 

and,  taking  the  tensors, 

T(y«^  +  y/5y  +  Vya)  =  ^JLl^L  =  2  area  base. 

But  Va/3  +  V/?y  +  Vya  +  V/?a  +  Vy^  +  Vay  =  0,  iu  which  the 
last  terms  are  twice  the  vector  areas  of  the  plane  fiices.  The 
sum  of  the  vector  areas  of  all  the  faces  is  therefore  zero.  Since 
an}-  pol3'edron  may  be  divided  into  tetraedra  by.plane  sections, 
whose  vector  areas  will  have  the  same  numerical  coefficient,  but 
have  opi)osite  signs  two  and  two,  the  sum  of  the  vector  areas  of 
any  polyedron  is  zero.  These  vector  areas  represent  the  pres- 
sures on  the  faces  of  a  polyedron  immersed  in  a  perfect  fluid 
subjected  to  no  external  forces.  For  rotation,  since  the  points 
of  application  of  these  pressures  are  the  centers  of  gravity  of 
the  faces,  to  which  the  vectors  are 

i(a+^  +  y),      i(/3  +  a),      Uy  +  fS),      i(a  +  7), 

we  have  the  couples 

--JVS(a  +  /8+y)(Va/3+V^y  +  Vya)-f-(a  +  /3)V^a+(/3  4-y) 
yyi3  +  (y  +  a)Vayj 
=  -lV(aV/Sy  +  y8Vya  +  yVa)8), 

since  aVa^  +  aVy8a  =  0,  etc.  But,  Equation  (123),  this  sum  is 
zero.     Hence  there  is  no  rotation. 


MISCELLANEOUS   EXAMPLES.  231 

100.   Miscellaneous  Examples. 

1.  lu  Fig.  58,  F,  A  aud  k  are  colliuear. 

2.  lu  Fig.   58,  AD"  — AE^  =  AB-  — AC^. 

3.  lu  Fig.  13,  if  the  lines  from  the  vertices  of  the  parallelo- 

gram through  o  aud  p  are  augle-bisectors,  omiip  is  a 
rectangle. 

4.  If  the  corresponding  sides  of  two  triangles  are  in  the  same 

ratio,  the  triangles  are  similar. 

5.  ^,  a,  y  being  the  vector  sides  of  a  plane  triangle,  if  (3—a-{-y, 

show  that  &-=c-— ca cos B+a6  cose. 

6.  The  sides  bc,  ca,  ab  of  a  triangle  are  produced  to  d,  e,  f, 

so  that  CD  =  7/iBC,  AE  =  ?JCA,  BF  =pAB.  Find  the  inter- 
sections Qi,  Qo,  Q3  of  EB,  FC  ;    FC,  DA  ;    DA,  EB. 

7.  In  any  right-angled  triangle,  four  times  the  sum  of  the 

squares  of  the  medials  to  the  sides  about  the  right  angle 
is  equal  to  five  times  the  square  of  the  hypothenuse. 

8.  If  ABC  be  any  triangle,  m  its  mean  point,  and  o  any  point 

in  space,  then 

AB-4-  BC-+  OA-  =  3(OA-+  OB-+  OC-)  —  (3  Om)^. 

9.  If  ABCD  be  any  quadrilateral,  m  its  mean  point,  and  o  any 

point  in  space,  tlieu 

AB-+  BC-+  C'D-+  DA- 

=  4(OA-+  OB-+OC--j-OD^)  —  (4om)^— AC^— BD-. 

10.  If  ABC  be  any  triangle,  and  c',  b',  a'  the  middle  points  of 

AB,  AC,  CB,  then,  o  being  any  point  in  space, 

AB-  +  BC-  +  CA-  =  4(0A-  +  0B-  +  0C-)— 4(ob'-+OC'--|-Oa'-). 

11.  If  ABC  be  any  triangle  and  51  its  mean  point,  then 

AB-+  BC--f  CA-  =  3  (aM-+  BM-4-  CM^)  . 

12.  Points  p,  Q,  R,  s  are  taken  in  the  sides  ab,  bc,  od,  da  of  a 

parallelogram,  so  that  ap  =  7?iab,  bq  =  ?nBC,  etc.  Show 
that  PQRS  is  a  parallelogram  whose  mean  point  coincides 
with  that  of  abcd. 


232  QUATERNIONS. 

13.  The  sides  of  any  quadrilateral  are  divided  equably  at  p,  q, 

R,  s,  and  the  points  of  division  joined  in  succession.  If 
PQKS  is  a  parallelogram,  the  original  quadrilateral  is  a 
parallelogram, 

14.  The  middle  points  of  the  three  diagonals  of  a  complete 

quadrilateral  arc  coUinear. 

15.  If  any  quadrilateral  be  divided  into  two  quadrilaterals  by 

any  cutting  line,  the  centers  of  the  three  are  collinear. 

16.  If  a  circle  be  described  about  the  mean  point  of  a  paral- 

lelogram as  a  center,  the  sum  of  the  squares  of  the  lines 
drawn  from  any  point  in  its  circumference  to  the  four 
angular  points  of  the  parallelogram  is  constant. 

17.  A  quadrilateral  possesses  the  following  property  :  any  point 

being  taken,  and  four  triangles  formed  by  joining  this 
point  with  the  angular  points  of  the  figure,  the  centers 
of  gi-avity  of  these  triangles  lie  in  the  circumference  of  a 
circle.  Prove  that  the  diagonals  of  this  quadrilateral  are 
at  right  angles  to  each  other. 

18.  The  sum  of  the  vector  perpendiculars  from  a,  b,  c, on 

any  line  through  their  mean  point  is  zero. 

19.  a,  b,  c  are  the  three  adjacent  edges  of  a  rectangular  paral- 

lelopiped.  Show  that  the  area  of  the  triangle  formed  by 
joining  their  extremities  is  ^V6"(r+  crcr+a-b-. 

20.  Given  the  co-ordinates  of  a,  b,  c,  d  referred  to  rectangular 

axes.  Find  the  volume  of  the  pyramid  o— abcd,  o  being 
the  origin. 

21.  Any  plane  through  the  middle  points  of  two  opposite  edges 

of  a  tetraedron  bisects  the  latter. 

22.  The  chord  of  contact  of  two  tangents  to  a  circle  drawn 

from  the  same  point  is  perpendicular  to  the  line  joining 
that  point  with  the  center. 

23.  If  two  circles  cut  each  other  and  from  one  point  of  section 

a  diameter  be  drawn  to  each  circle,  the  line  joining  then- 
extremities  is  parallel  to  the  line  joining  their  centers, 
and  passes  through  the  other  point  of  section. 


MISCELLANEOUS   EXAMPLES.  233 

24.  The  square  of  the  sum  of  the  diameters  of  two  circles,  tan- 

gent at  a  common  point,  is  equal  to  the  sum  of  the 
squares  of  any  two  common  chords  through  the  point  of 
tangency,  at  right  angles  to  each  other. 

25.  T  is  any  point  without  a  circle  whose  centre  is  c  ;    from  t 

draw  two  tangents  tp,  tq,  also  any  line  cutting  the  circle 
in  V,  and  pq  in  r  ;  draw  cs  perpendicular  to  tv.     Then 

SR  .  ST  =  SV^. 

26.  If  a  series  of  circles,  tangent  at  a  common  point,  are  cut 

by  a  fixed  circle,  the  lines  of  section  meet  in  a  point. 

27.  In  Ex.  26,  the  intersections  of  the  pairs  of  tangents  to  the 

fixed  circle,  at  the  points  of  section,  lie  in  a  straight 
line. 

28.  If  three  given  circles  are  cut  by  any  circle,  the  lines  of 

section  form  a  triangle,  the  loci  of  whose  angular  points 
are  right  lines  perpendicular  to  the  lines  joining  the 
centers  of  the  given  circles. 

29.  The  three  loci  of  Ex.  28  meet  in  a  point. 

30.  Given  the  base  of  an  isosceles  triangle,  to  find  the  locus  of 

the  vertex. 

31.  Find  the  locus  of  the  center  of  a  circle  which  passes  through 

two  given  points. 

32.  Find  the  locus  of  the  center  of  a  sphere  of  given  radius, 

tangent  to  a  given  sphere. 

33.  The  locus  of  the  point  from  which  two  circles   subtend 

equal  angles  is  a  circle,  or  a  right  line. 

34.  Given  the  base  of  a  triangle,  and  7n  times  the  square  of 

one  side  plus  w  times  the  square  of  the  other,  to  find  the 
locus  of  the  vertex. 

35.  Given  the  base  and  the  sum  of  the  squares  of  the  sides  of 

a  triangle,  to  find  the  locus  of  the  vertex. 

36.  In  Ex.  35,  given  the  difference  of  the  squares,  to  find  the 

locus. 


234  QUATERNIONS. 

37.  OB  and  oa  are  any  two  lines,  and  mp  is  a  line  parallel  to 

OB.     Find  the  locus  of  the  intersection  of  oq  and  bq 
drawn  parallel  to  ap  and  op,  respectively. 

38.  From  a  fixed  point  p,  on  the  surface  of  a  sphere,  chords 

pp',  pp", are  drawn.     Find  the  locus  of  a  point  o  on 

these  chords,  such  that  pp'.  po  =  m'. 

39.  A  line  of  constant  length  moves  witli  its  extremities  on  two 

straight  lines  at  right  angles  to  each  other.     Find  the 
locus  of  its  middle  point. 

40.  Find  the  locus  of  a  point  such  that  if  straight  lines  be 

drawn  to  it  from  the  four  corners  of  a  square,  the  sum 
of  their  squares  is  constant. 

41.  Find  the  locus  of  a  point  the  square  of  whose  distance 

from  a  given  point  is  proportional  to  its  distance  from  a 
given  line. 

42.  Find  the  locus  of  the  feet  of  perpendiculars  from  the  origin 

on   planes  cutting  off  pj'ramids  of  equal  volume  from 
three  rectangular  co-ordinate  axes. 

43.  Given  the  base  of  a  triangle  and  the  ratio  of  the  sides,  to 

find  the  locus  of  the  vertex. 

44.  Show  that  TapYp/?  =  (Ja/3)-  is  the  equation  of  a  hyperbola 

whose  asymptotes  are  parallel  to  a  and  /8. 

45.  Find  the  point  on  an  ellipse  the  tangent  to  which  cuts  off 

equal  distances  on  the  axes. 
4G.  A  and  b  are  two  similar,  similarly  situated,  and  concentric 
ellipses  ;  c  is  a  third  ellipse  similar  to  a  and  b,  its  center 
being  on  the  circumference  of  b,  and  its  axes  parallel  to 
those  of  A  and  b  :  show  that  the  chord  of  intersection  of 
A  and  B  is  parallel  to  the  tangent  to  u  at  the  center  of  c. 


Presswork  by  Ginn  &  Co.,  Boston. 


MATHEMATICS.  165 


Wentworth  &  Hill's  Exercises  in  Algebra. 

I.  Exercise  Manual.     i2mo.     Boards.      232  pages.      Mailing  price, 

40  cts. ;  Introduction  price,  35  cts.  —  II.  EXAMINATION  Manual.    i2mo. 

Boards.     159  pages.      Mailing  price,  40  cts. ;  Introduction  price,  35  cts. 

Both  in  one  volume,  70  cts.     Answers  to  both  parts  together,  25  cts. 

The  first  part  (Exercise  Manual)  contains  about  4500  problems 
c/assified  and  arranged  according  to  the  usual  order  of  text-books 
in  Algebra ;  and  the  second  part  (Examination  Manual)  contains 
nearly  300  examination-papers,  progressive  in  character,  and  well 
adapted  to  cultivate  skill  and  rapidity  in  solving  i^roblems. 

Wentworth  &  Hill's  Exercises  in  Arithmetic. 

I.  Exercise  Manual.     II.  Examination  Manual.     i2mo.     Boards. 

148  pages.     Mailing  price,  40  cts.;  Introduction  price,  35  cts.     Both  in 

one  volume,  70  cts.     Anszvers  to  both  parts  together,  25  cts. 

The  first  part  (Exercise  Manual)  contains  problems  for  daily 
practice,  classified  and  arranged  in  the  common  order;  and  the 
second  part  (Examination  Manual)  contains  300  examination-papers, 
progressive  in  character.  The  second  part  has  already  been  issued? 
and  the  first  part  will  be  ready  in  August,  18S6. 

Analytic  Geometry. 

By  G.  A.   Wentworth.     i2mo.     Half  morocco.     000   pp.     Mailing 
price,  $0.00;  for  Introduction, 

The  aim  of  this  work  is  to  present  the  elementary  parts  of  the 
subject  in  the  best  form  for  class-room  use. 

The  connection  between  a  locus  and  its  equation  is  made  perfectly 
clear  in  the  opening  chapter. 

The  exercises  are  well  graded  and  designed  to  secure  the  best 
mental  training. 

By  adding  a  supplement  to  each  chapter  provision  is  made  for  a 
shorter  or  more  extended  course,  as  the  time  given  to  the  subject 
will  permit. 

The  book  is  divided  into  chapters  as  follows :  — 
Chapter        I.     Loci  and  their  Equations. 
"    '        II.     The  Straight  Line. 
"  III.     The  Circle. 

"  IV.     Different  Systems  of  Co-ordinates, 

"  V.     The  Parabola. 

"  VI.     The  Ellipse. 

"         VII.     The  Hyperbola. 
"       VIII.     The  General  Equation  of  the  Second  Degree. 


1 SS  MA  THE  MA  TICS. 


Peirce's  Three  and  Four  Place  Tables  of  Loga- 

rithmic  and  Trigonometric  Functions.  By  James  Mills  Peikce, 
University  Professor  of  Mathematics  in  Harvard  University.  Quarto. 
Cloth.     Mailing  Price,  45  cts. ;  Introduction,  40  cts. 

Four-place  tables  require,  in  the  long  run,  only  half  as  much  time 
"iS  five-place  tables,  one-third  as  much  time  as  six-place  tables,  and 
one-fourth  as  much  as  those  of  seven  places.  They  are  sufficient 
for  the  ordinary  calculations  of  Surveying,  Civil,  Mechanical,  and 
Mining  Engineering,  and  Navigation ;  for  the  work  of  the  Physical 
or  Chemical  Laboratory,  and  even  for  many  computations  of  Astron- 
omy. They  are  also  especially  suited  to  be  used  in  teaching,  as  they 
illustrate  principles  as  well  as  the  larger  tables,  and  with  far  less 
expenditure  of  time.  The  present  compilation  has  been  prepared 
with  care,  and  is  handsomely  and  clearly  printed. 

Elements  of  the  Differential  Calculus. 

With  Numerous  Examples  and  Applications.  Designed  for  Use  as  a 
College  Text-Book.  I5y  W.  E.  Bvekly,  Professor  of  Mathematics, 
Harvard  University.  8vo.  273  pages.  Mailing  Price,  $2.15  ;  Intro- 
duction, S2.00. 

This  book  embodies  the  results  of  the  author's  experience  in 
teaching  the  Calculus  at  Cornell  and  Harvard  Universities,  and  is 
intended  for  a  text-book,  and  not  for  an  exhaustive  treatise.  Its 
peculiarities  are  the  rigorous  use  of  the  Doctrine  of  Limits,  as  a 
foundation  of  the  subject,  and  as  preliminary  to  the  adoption  of  the 
more  direct  and  practically  convenient  infinitesimal  notation  and 
nomenclature ;  the  early  introduction  of  a  few  simple  formulas  and 
methods  for  integrating  ;  a  rather  elaborate  treatment  of  the  use  of 
infinitesimals  in  pure  geometry ;  and  the  attempt  to  excite  and  keep 
up  the  interest  of  the  student  by  bringing  in  throughout  the  whole 
book,  and  not  merely  at  the  end,  numerous  applications  to  practical 
problems  in  geometry  and  mechanics. 


James  Mills  Peirce,  Prof,  of 
Math.,  Harvard  Univ.  (From  the  Har- 
vard Register') :  In  mathematics,  as  in 
other  branches   of  study,  the  need  is 


is  general  without  being  superficial; 
limited  to  leading  topics,  and  yet  with- 
in its  limits;  thorough,  accurate,  and 
practical ;  adapted  to  the  communica- 


now  very  much  felt  of  teaching  which '  tion  of  some  degree  of  power,  as  well 


MATHEMATICS. 


189 


as  knowledge,  but  free  from  details 
which  are  important  only  to  the  spe- 
cialist. Professor  Byerly's  Calculus 
appears  to  be  designed  to  meet  this 
want.  .  .  .  Such  a  plan  leaves  much 
room  for  the  exercise  of  individual 
judgment ;  and  differences  of  opinion 
will  undoubtedly  exist  in  regard  to  one 
and  another  point  of  this  book.  But 
all  teachers  will  agree  that  in  selection, 
arrangement,  and  treatment,  it  is,  on 
the  whole,  in  a  very  high  degree,  wise, 
able,  marked  by  a  true  scientific  spirit, 
and  calculated  to  develop  the  same 
spirit  in  the  learner.  .  .  .  The  book 
contains,  perhaps,  all  of  the  integral 
calculus,  as  well  as  of  the  differential, 
that  is  necessary  to  the  ordinary  stu- 
dent. And  with  so  much  of  this  great 
scientific  method,  every  thorough  stu- 
dent of  physics,  and  every  general 
scholar  who  feels  any  interest  in  the 
relations  of  abstract  thought,  and  is 
capable  of  grasping  a  mathematical 
idea,  ought  to  be  familiar.  One  who 
aspires  to  technical  learning  must  sup- 
plement his  mastery  of  the  elements 
by  the  study  of  the  comprehensive 
theoretical  treatises.  .  .  .  But  he  who  is 
thoroughly  acquainted  with  the  book 
before  us  has  made  a  long  stride  into 
a  sound  and  practical  knowledge  of 
the  subject  of  the  calculus.  He  has 
begun  to  be  a  real  analyst. 

H.  A.  Ne-wi;on,  Prof,  of  Math,  in 
Yale  Coll.,  New  Haven  :  I  have  looked 
it  through  with  care,  and  find  the  sub- 
ject very  clearly  and  logically  devel- 
oped. I  am  strongly  inclined  to  use  it 
in  my  class  next  year. 

S.  Hart,  recent  Prof,  of  Math,  in 
Trinity  Coll.,  Conn. :  The  student  can 
hardly  fail,  I  think,  to  get  from  the  book 
an  exact,  and,  at  the  same  time,  a  satis- 
factory explanation  of  the  principles  on 
which  the  Calculus  is  based;  and  the 
introduction  of  the  simpler  methods  of 


integration,  as  they  are  needed,  enables 
applications  of  those  principles  to  be 
introduced  in  such  a  way  as  to  be  both 
interesting  and  instructive. 

Charles  Kraus,  Techniker,  Pard- 
tibitz,  Bohemia,  Austria  :  Indem  ich 
den  Empfang  Ihres  Buches  dankend 
bestaetige  muss  ich  Ihnen,  hoch  geehr- 
ter  Herr  gestehen,  dass  mich  dasselbe 
sehr  erfreut  hat,  da  es  sich  durch 
grosse  Reichhaltigkeif,besonders  klare 
Schreibweise  und  vorzuegliche  Behand- 
lung  des  Stoffes  auszeichnet,  und  er- 
weist  sich  dieses  Werk  als  eine  bedeut- 
ende  Bereicherung  der  mathematischen 
Wissenschaft. 

De  Volsoa  Wood,  Prof,  of 
Math.,  Stevens'  Inst.,  Hoboken,  N.f. : 
To  say,  as  I  do,  that  it  is  a  first-class 
work,  is  probably  repeating  what  many 
have  already  said  for  it.  I  admire  the 
rigid  logical  character  of  the  work, 
and  am  gratified  to  see  that  so  able  a 
writer  has  shown  explicitly  the  relation 
between  Derivatives,  Infinitesimals,  and 
Differentials.  The  method  of  Limits 
is  the  true  one  on  which  to  found  the 
science  of  the  calculus.  The  work  is 
not  only  comprehensive,  but  no  vague- 
ness is  allowed  in  regard  to  definitions 
or  fundamental  principles. 

Del    Kemper,    Prof   of  Math., 

Hampden  Sidney  CoU.,  Fa. :  My  high 
estimate  of  it  has  been  amply  vindi- 
cated by  its  use  in  the  class-room. 

R.  H.  Graves,  Prof,  of  Math., 
Univ.  of  North  Carolina :  I  have  al- 
ready decided  to  use  it  with  my  next 
class ;  it  suits  my  purpose  better  than 
anv  other  book  on  the  same  subject 
with  which  I  am  acquainted. 

Edw.  Brooks,  Author  of  a  Series 

of  Math.  :  Its  statements  are  clear  and 
scholarly,  and  its  methods  thoroughly 
analytic  and  in  the  spirit  of  the  latest 
mathematical  thought. 


190 


MA  THEM  A  TICS. 


Syllabus  of  a  Course  in  Plane  Trigonometry. 

By  W.  E.  Byerly.     8vo.     8  pages.     Mailing  Price,  lo  cts. 

Syllabus  of  a  Course  in  Plane  Analytical  Geom- 

elry.     By  W.  E.  Bykkly.     8vo.     12  pages,     ilailing  Price,  10  cts. 

Syllabus  of  a  Course  in  Plane  Analytic  Geom- 

ctry     (^Advauced  Course.)     By   W.    E.   Byerly,   Professor  of  Mathe- 
matics, Plarvard  University.     8vo.     12  pages.     Mailing  Price,  10  cts. 

Syllabus  of  a  Course  in  Analytical  Geometry  of 

Three  Dimensioyis.     By  W.  E.  Byekly.     Svo.      10  pages.      Mailing 
Price,  10  cts. 

Syllabus   of  a   Course  on  Modern  Methods   in 

Analytic  Geometry.     By  W.  E.  Byekly.      8vo.      8  pages.      Mailing 
Price,  10  cts. 

Syllabus  of  a  Course  in  the  Theory  of  Equations. 

By  W.  E.  Byerly.     8vo.     8  pages.     Mailing  Price,  10  cts. 

Elements  of  the  Integral  Calculus. 

By  W.  E.  Byerly,  Professor  of  Mathematics  in  Harvard  University. 
8vo.     204  pages.     Mailing  Price,  $2.15;  Introduction,  $2.00. 

This  volume  is  a  sequel  to  the  author's  treatise  on  the  Differential 
Calculus  (see  page  134),  and,  like  that,  is  written  as  a  text-book. 
The  last  chapter,  however,  —  a  Key  to  the  Solution  of  Differential 
Equations,  —  may  prove  of  service  to  working  mathematicians. 


H.  A.  Newton,  Pro/,  of  Math., 
Yale  Coll. :  We  shall  use  it  in  my 
optional  class  next  term. 

Mathematical  Visitor :  The 
subject  is  presented  very  clearly.  It  is 
the  first  American  treatise  on  the  Cal- 
culus that  we  have  seen  which  devotes 
any  space  to  average  and  probability. 
\ 

Schoolmaster,  London :  The 
merits  of  this  work  are  as  marked  as 


those  of  the  Differential   Calculus  by 
the  same  author. 

Zion's  Herald  :  A  text-book  every 
way  worthy  of  the  venerable  University 
in  which  the  author  is  an  honored 
teacher.  Cambridge  in  Massachusetts, 
like  Cambridge  in  England,  preserves 
its  reputation  for  the  breadth  and  strict- 
ness of  its  mathematical  requisitions, 
and  these  form  the  spinal  column  of  a 
liberal  education. 


192 


MA  THE  MA  TICS. 


Elements  of  the  Differential  and  Integral  Calculus. 


With  Examples  and  Applications.  By  J. 
Mathematics  in  Madison  University.  8vo. 
price,  $1.95;    Introduction  price,  ^1.80. 


M.  Taylor,  Professor  of 
Cloth.     249  pp.     Mailing 


The  aim  of  this  treatise  is  to  present  simply  and  concisely  the 
fundamental  problems  of  the  Calculus,  their  solution,  and  more 
common  applications.  Its  axiomatic  datum  is  that  the  change  of  a 
variable,  when  not  uniform,  may  be  conceived  as  becoming  uniform 
at  any  value  of  the  variable. 

It  employs  the  conception  of  rates,  which  affords  finite  differen- 
tials, and  also  the  simplest  and  most  natural  view  of  the  problem  of 
the  Differential  Calculus.  This  problem  of  finding  the  relative 
rates  of  change  of  related  variables  is  afterwards  reduced  to  that  of 
finding  the  limit  of  the  ratio  of  their  simultaneous  increments  ;  and, 
in  a  final  chapter,  the  latter  problem  is  solved  by  the  principles  of 
infinitesimals. 

Many  theorems  are  proved  both  by  the  method  of  rates  and  that 
of  limits,  and  thus  each  is  made  to  throw  light  upon  the  other. 
The  chapter  on  differentiation  is  followed  by  one  on  direct  integra- 
tion and  its  more  important  applications.  Throughout  the  work 
there  are  numerous  practical  problems  in  Geometry  and  Mechanics, 
which  serve  to  exhibit  the  power  and  use  of  the  science,  and  to 
excite  and  keep  alive  the  interest  of  the  student. 

Judging  from  the  author's  experience  in  teaching  the  subject,  it 
is  believed  that  this  elementary  treatise  so  sets  forth  and  illustrates 
the  highly  practical  nature  ef  the  Calculus,  as  to  awaken  a  lively 
interest  in  many  readers  to  whom  a  more  abstract  method  of  treat- 
ment would  be  distasteful. 


Oren  Root,  Jr.,  Prof,  of  Math., 
Ha^nilton  Coll.,  N.Y.:  In  reading  the 
manuscript  I  was  impressed  by  the 
clearness  of  definition  and  demonstra- 
tion, the  pertinence  of  illustration,  and 
the  happy  union  of  exclusion  and  con- 
densation. It  seems  to  me  most  admir- 
ably suited  for  use  in  college  classes. 
I  prove  my  regard  by  adopting  this  as 
our  text-book  on  the  calculus. 


C.    M.    Charrappin,    S.J.,   St. 

Louis  Univ. ;  I  have  given  the  book  a 
thorough  examination,  and  I  am  satis- 
fied that  it  is  the  best  work  on  the  sub- 
ject I  have  seen.  I  mean  the  best 
work  for  what  it  was  intended. — a  text- 
book. I  would  like  very  much  to  in- 
troduce it  in  the  University. 
(fa7t.  12,  1885.) 


194  MA  THEM  A  TICS. 


Metrical  Geometry:  An  Elementary  Treatise  on 

Mensuration.  By  George  Bruce  Halsted,  Ph.D.,  Prof.  Mathema- 
tics, University  of  Texas,  Austin.  i2mo.  Cloth.  246  pages.  Mailing 
price,  $l.lo;   Introduction,  $1.00. 

This  work  applies  new  principles  and  methods  to  simplify  the 
measurement  of  lengths,  angles,  areas,  and  volumes.  It  is  strictly 
demonstrative,  but  uses  no  Trigonometry,  and  is  adapted  to  be  taken 
up  in  connection  with,  or  following  any  elementary  Geometry.  It 
treats  of  accessible  and  inaccessible  straight  lines,  and  of  their  inter- 
dependence when  in  triangles,  circles,  etc. ;  also  gives  a  more  rigid 
rectification  of  the  circumference,  etc.  It  introduces  the  natural 
unit  of  angle,  and  deduces  the  ordinary  and  circular  measure. 
Enlisting  the  auxiliary  powers  which  modern  geometers  have  recog- 
nized in  notation,  it  binds  up  each  theorem  also  in  a  self-e.xplanatory 
formula,  and  this  throughout  the  whole  book  on  a  system  which 
renders  confusion  impossible,  and  surprisingly  facilitates  acquire- 
ment, as  has  been  tested  with  very  large  classes  in  Princeton  College. 
In  addition  to  all  the  common  propositions  about  areas,  a  new 
method,  applicable  to  any  polygon,  is  introduced,  which  so  simplifies 
and  shortens  all  calculations,  that  it  is  destined  to  be  universally 
adopted  in  surveying,  etc.  In  addition  to  the  circle,  sector,  segment, 
zone,  annulus,  etc.,  the  parabola  and  ellipse  are  measured ;  and  be- 
sides the  common  broken  and  curved  surfaces,  the  theorems  of 
Pappus  are  demonstrated.  Especial  mention  should  be  made  of  the 
treatment  of  solid  angles,  which  is  original,  introducing  for  the  first 
time,  we  think,  the  natural  unit  of  solid  angle,  and  making  spherics 
easy.  For  solids,  a  single  informing  idea  is  fixed  upon  of  such 
fecundity  as  to  place  within  the  reach  of  children  results  heretofore 
only  given  by  Integral  Calculus.  Throughout,  a  hundred  illustrative 
examples  are  worked  out,  and  at  the  end  are  five  hundred  carefully 
arranged  and  indexed  exercises,  using  the  metric  system. 

OPINIONS. 


Simon  Newcomb,  Nautical  Al- 
manac Office,  Washington,  D.C.:  iSm 
much  interested  in  your  Mensuration, 
and  wish  I  had  seen  it  in  time  to  have 


Alexander  MacFarlane,  Exam- 
iner in  Mathematics  to  the  University 
of  Edinburgh,  Scotland :  The  method, 
figures,  and  examples  appear  excellent, 


some  exercises  suggested  by  it  put  into    and  I  anticipate  much  benefit  from  its 
my  Geometry.     {Sept.  8,  1881.)  I  minute  perusal. 


MA  THEM  A  TICS.  195 


Elementary  Co-ordinate  Geometry. 

By  W.  B.  Smith,  Professor  of  Physics,  Missouri  State  University.     l2mo. 
Cloth.     312  pp.     Mailing  price,  ^2.15;   for  Introduction,  32.00. 

While  in  the  study  of  Analytic  Geometry  either  gain  of  knowledge 
or  culture  of  mind  may  be  sought,  the  latter  object  alone  can  justify 
placing  it  in  a  college  curriculum.  Yet  the  subject  may  be  so  pur- 
sued as  to  be  of  no  great  educational  value.  Mere  calculation,  or  the 
solution  of  problems  by  algebraic  processes,  is  a  very  inferior  dis- 
cipline of  reason.  Even  geometry  is  not  the  best  discipline.  In  all 
thinking  the  real  difficulty  lies  in  forming  clear  notions  of  things. 
In  doing  this  all  the  higher  faculties  are  brought  into  play.  It  is  this 
formation  of  concepts,  therefore,  that  is  the  essential  part  of  mental 
training.  He  who  forms  them  clearly  and  accurately  may  be  safely 
trusted  to  put  them  together  correctly.  Nearly  every  seeming  mis- 
take in  reasoniiig  is  really  a  mistake  in  conception. 

Such  considerations  have  guided  the  composition  of  this  book. 
Concepts  have  been  introduced  in  abundance,  and  the  proofs  made 
to  hinge  directly  upon  them.  Treated  in  this  way  the  subject 
seems  adapted,  as  hardly  any  other,  to  develop  the  power  of 
thought. 

Some  of  the  special  features  of  the  work  are :  — 

1.  Its  SIZE  is  such  it  can  be  mastered  in  the  time  generally 
allowed. 

2.  The  SCOPE  is  far  wider  than  in  any  other  American  work. 

3.  The  combination  of  small  size  and  large  scope  has  been  secured 
through  SUPERIOR  METHODS,  —  7>wdern,  direct,  and  rapid. 

4.  Conspicuous  among  such  methods  is  that  of  determinants, 
here  presented,  by  the  union  of  theory  and  practice,  in  its  real 
power  and  beauty. 

5.  Confusion   is   shut   out  by  a  consistent    and    self-explaining 

NOTATION. 

6.  The  ORDER  OF  development  is  natural,  and  leads  without 
break  or  turn  from  the  simplest  to  the  most  complex.  The  method 
is  heuristic. 

7.  The  student's  grasp  is  strengthened  by  numerous  exercises. 

8.  The  work  has  been  tested  at  every  point  in  the  class- 
room. 


196  MA  THEM  A  TICS. 


Determinants. 

The  Theory  of  Determinants:  an  Elementary  Treatise.  By  Paul  H. 
Hani.s,  U.S.,  Professor  of  Mathematics  in  the  University  of  Colorado. 
Svo.    Cloth,    ooo  pages.    Mailing  price,  3o.oo;   for  Introduction,  $o.oo. 

This  is  a  text-book  for  the  use  of  students  in  colleges  and  tech- 
nical schools.  The  need  of  an  American  work  on  determinants 
has  long  been  felt  by  all  teachers  and  students  who  have  extended 
their  reading  beyond  the  elements  of  mathematics.  The  importance 
of  the  subject  is  no  longer  overlooked.  The  shortness  and  elegance 
imparted  to  many  otherwise  tedious  processes,  by  the  introduction 
of  determinants,  recommend  their  use  even  in  the  more  elementary 
branches,  while  the  advanced  student  cannot  dispense  with  a  knowl- 
edge of  these  valuable  instruments  of  research.  Moreover,  deter- 
minants are  employed  by  all  modern  writers. 

This  book  is  written  especially  for  those  who  have  had  no  previous 
knowledge  of  the  subject,  and  is  therefore  adapted  to  self-instruction 
as  well  as  to  the  needs  of  the  class-room.  To  this  end  the  subject 
is  at  first  presented  in  a  very  siinple  manner.  As  the  reader  ad- 
vances, less  and  less  attention  is  given  to  details.  Throughout  the 
entire  work  it  is  the  constant  aim  to  arouse  and  enliven  the  reader's 
interest  by  first  showing  how  the  various  concepts  have  arisen 
naturally,  and  by  giving  such  applications  as  can  be  presented  with- 
out exceeding  the  limits  of  the  treatise.  The  work  is  sufficiently 
comprehensive  to  enable  the  stiident  that  has  mastered  the  volume 
to  use  the  determinant  notation  with  ease,  and  to  pursue  his  further 
reading  in  the  modern  higher  algebra  with  pleasure  and  profit. 

In  Chapter  I.  the  evolution  of  a  theory  of  determinants  is  touched 
upon,  and  it  is  shown  how  determinants  are  produced  in  the  process 
of  eliminating  the  variables  from  systems  of  simple  equations  with 
some  further  preliminary  notions  and  definitions. 

In  Chapter  II.  the  most  important  properties  of  determinants  are 
discussed.  Numerous  examples  serve  to  fix  and  exemplify  the  prin- 
ciples deduced. 

Chapter  III.  comprises  half  the  entire  volume.  It  is  the  design 
of  this  chapter  to  familiarize  the  reader  with  the  most  important 
special  forms  that  occur  in  application,  and  to  enable  him  to  realize 
the  practical  usefulness  of  determinants  as  instruments  of  research. 

\Ready  yune  i. 


MA  THEM  A  TICS.  197 


Examples  of  Differentia/  Equations. 

By  George  A.  Osborne,  Professor  of  Mathematics  in  the  INIassachusetts 
Institute  of  Technology,  Boston.  i2mo.  Cloth,  viii  +  50  pp.  Mail- 
ing price,  60  cts.;   for  Introduction,  50  cts. 

Notwithstanding  the  importance  of  the  study  of  Differential  Equa- 
tions, either  as  a  branch  of  pure  mathematics,  or  as  applied  to 
Geometry  or  Physics,  no  American  work  on  this  subject  has  been 
published  containing  a  classified  series  of  examples.  This  book  is 
intended  to  supply  this  want,  and  provides  a  series  of  nearly  three 
hundred  examples  with  answers  systematically  arranged  and  grouped 
under  the  different  cases,  and  accompanied  by  concise  rules  for  the 
solution  of  each  case. 

It  is  hoped  that  the  work  will  be  found  useful,  not  only  in  the 
study  of  this  important  subject,  but  also  by  way  of  reference  to 
mathematical  students  generally  whenever  the  solution  of  a  differen- 
tial equation  is  required. 

Elements  of  t/ie  Theory  of  tiie  Newtonian  Poten- 

tial  Function.  By  B.  O.  Peirce,  Assistant  Professor  of  Mathematics 
and  Physics,  Harvard  University.  i2mo.  Cloth.  154  pages.  Mailing 
price,  5l-6o;       for  Introduction,  $1.50. 

A  knowledge  of  the  properties  of  this  function  is  essential  for 
electrical  engineers,  for  students  of  mathematical  physics,  and  for 
all  who  desire  more  than  an  elementary  knowledge  of  experimental 
physics. 

This  book,  based  upon  notes  made  for  class-room  use,  was  written 
because  no  book  in  English  gave  in  simple  form,  for  the  use  of 
students  who  know  something  of  the  calculus,  so  much  of  the  theory 
of  the  potential  function  as  is  needed  in  the  study  of  physics. 
Both  matter  and  arrangement  have  been  practically  adapted  to  the 
end  in  view. 

Chapter  I.  The  Attraction  of  Gravitation. 

II.  The  Newtonian  Potential  Function  in  the  Case  of  Gravitation. 

III.  The  Newtonian  Potential  Function  in  the  Case  of  Repulsive 

Forces. 

IV.  Surface  Distributions.      Green's  Theorem. 

V.  Application    of   the    Results    of   the    Preceding  Chapters    to 
Electrostatics. 


Mathematics. 

Introd. 
Prices. 

Byerly Differential  Calculus §2.00 

lutegral  Calculus 2.00 

Ginn Addition  Manual 15 

Halsted Mensuration 1.00 

Hardy    Quaternions 2.00 

Hill Geometry  for  Beginners    1.00 

Sprague Rapid  Addition 10 

Taylor   Elements  of  the  Calculus 1.80 

Wentworth    . Grammar  School  Aritlunetic    75 

Shorter  Course  in  Algebra 1.00 

Elements  of  Algebra 1.12 

Complete  Algebra  1.40 

Plane  Geometry   75 

Plane  and  Solid  Geometry  1.25 

Plane  and  Solid  Geometry,  and  Trigonometry  1.40 

Plane  Trigonometry  and  Tables.     Paper. .       .60 

PI.  and  Sph.  Trig.,  Surv.,  and  Navigation   .    1.12 

PL  and  Sph.  Trig.,  Surv.,  and  Tables 1.25 

Trigonometric  Formulas 1.00 

Wentworth  &  Hill :  Practical  Arithmetic 1.00 

Abridged  Practical  Arithmetic 75 

Exercises  in  Arithmetic 

Part  I.  Exercise  Manual 

Part  II.  Examination  Manual 35 

Answers  (to  both  Parts) 25 

Exercises  in  Algebra 70 

Part  I.  Exercise  Mximial 35 

Part  II.  Examination  Manual 35 

Answers  (to  both  Parts)    25 

Exercises  in  Geometry 70 

Five-place  Log.  and  Trig.  Tables  (7  Tables)      .50 

Five-place  Log.  and  Trig.  Tables  (Cojh;)..£' J.)  1.00 

Wentworth  &  Reed :  First  Steps  in  Number,  Pupils'  Edition       .30 

Teachers'  Edition,  complete       .90 

Parts  I.,  II.,  and  III.  (separate),  each      .30 

Wheeler Plane  and  Spherical  Trig,  and  Tables 1.00 

Copies  sent  to  Teachers  for  examination,  with  a  view  to  Introduction, 

on  receipt  of  Introduction  Price. 


GINN  k  COMPANY,  Publisliers. 

BOSTON.  NEW   YORK.  CHICA( 


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